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Measurable cardinal Wikipedia Contents

1 Atom ( theory) 1 1.1 Definition ...... 1 1.2 Examples ...... 1 1.3 Non-atomic measures ...... 1 1.4 See also ...... 2 1.5 Notes ...... 2 1.6 References ...... 2

2 of 3 2.1 Types of game that are determined ...... 3 2.2 Incompatibility of the with the ...... 3 2.3 Infinite and the axiom of determinacy ...... 4 2.4 Large cardinals and the axiom of determinacy ...... 4 2.5 See also ...... 4 2.6 References ...... 5 2.7 Further reading ...... 5

3 6 3.1 History ...... 6 3.2 Motivation ...... 8 3.3 Formal definition ...... 9 3.4 Cardinal arithmetic ...... 10 3.4.1 ...... 10 3.4.2 Cardinal addition ...... 10 3.4.3 Cardinal multiplication ...... 11 3.4.4 Cardinal exponentiation ...... 11 3.5 The ...... 12 3.6 See also ...... 12 3.7 References ...... 12 3.8 External links ...... 13

4 ( theory) 14 4.1 Examples ...... 14

i ii CONTENTS

4.2 Paradoxes ...... 14 4.3 Classes in formal set theories ...... 14 4.4 References ...... 15 4.5 External links ...... 15

5 Club filter 16 5.1 References ...... 16

6 Club set 17 6.1 Formal definition ...... 17 6.2 The closed unbounded filter ...... 17 6.3 See also ...... 18 6.4 References ...... 18

7 Continuum hypothesis 19 7.1 of infinite sets ...... 19 7.2 Independence from ZFC ...... 20 7.3 for and against CH ...... 20 7.4 The generalized continuum hypothesis ...... 21 7.4.1 Implications of GCH for cardinal exponentiation ...... 22 7.5 See also ...... 22 7.6 References ...... 22 7.7 External links ...... 23

8 Critical point () 24 8.1 References ...... 24

9 Dana Scott 25 9.1 Early career ...... 25 9.2 University of California, Berkeley, 1960–1963 ...... 25 9.2.1 Modal and tense logic ...... 26 9.3 Stanford, Amsterdam and Princeton, 1963–1972 ...... 26 9.4 Oxford University, 1972–1981 ...... 26 9.4.1 Semantics of programming languages ...... 26 9.5 Carnegie Mellon University 1981–2003 ...... 27 9.6 See also ...... 27 9.7 Bibliography ...... 27 9.7.1 Works by Scott ...... 27 9.7.2 Other works ...... 27 9.8 References ...... 27 9.9 External links ...... 28

10 Diagonal intersection 29 10.1 See also ...... 29 CONTENTS iii

10.2 References ...... 29

11 Elementary equivalence 30 11.1 Elementarily equivalent structures ...... 30 11.2 Elementary substructures and elementary extensions ...... 30 11.3 Tarski–Vaught test ...... 31 11.4 Elementary ...... 31 11.5 References ...... 31

12 Equiconsistency 32 12.1 ...... 32 12.2 Consistency strength ...... 32 12.3 See also ...... 33 12.4 References ...... 33

13 Extender (set theory) 34 13.1 Formal definition of an extender ...... 34 13.2 Defining an extender from an elementary ...... 34 13.3 References ...... 35

14 Fodor’ lemma 36 14.1 Proof ...... 36 14.2 Fodor’s lemma for trees ...... 36 14.3 References ...... 36

15 Huge cardinal 37 15.1 Variants ...... 37 15.2 Consistency strength ...... 38 15.3 ω-huge cardinals ...... 38 15.4 See also ...... 38 15.5 References ...... 38

16 39 16.1 Models and consistency ...... 39 16.2 Existence of a proper class of inaccessibles ...... 40 16.3 α-inaccessible cardinals and hyper-inaccessible cardinals ...... 40 16.4 Two model-theoretic characterisations of inaccessibility ...... 40 16.5 See also ...... 41 16.6 References ...... 41

17 Ineffable cardinal 42 17.1 References ...... 42

18 Intersection (set theory) 43 iv CONTENTS

18.1 Basic definition ...... 43 18.1.1 Intersecting and disjoint sets ...... 45 18.2 Arbitrary intersections ...... 46 18.3 Nullary intersection ...... 47 18.4 See also ...... 48 18.5 References ...... 48 18.6 Further reading ...... 48 18.7 External links ...... 48

19 49 19.1 Definition ...... 49 19.1.1 Intuition ...... 49 19.2 Examples ...... 50 19.3 Properties ...... 50 19.4 Null sets ...... 51 19.5 Construction of the Lebesgue measure ...... 52 19.6 to other measures ...... 52 19.7 See also ...... 53 19.8 References ...... 53

20 54 20.1 Measurable ...... 54 20.2 Real-valued measurable ...... 55 20.3 See also ...... 55 20.4 References ...... 55

21 Measure (mathematics) 56 21.1 Definition ...... 56 21.2 Examples ...... 57 21.3 Properties ...... 57 21.3.1 Monotonicity ...... 58 21.3.2 Measures of infinite unions of measurable sets ...... 58 21.3.3 Measures of infinite intersections of measurable sets ...... 58 21.4 Sigma-finite measures ...... 58 21.5 Completeness ...... 59 21.6 Additivity ...... 59 21.7 Non-measurable sets ...... 59 21.8 Generalizations ...... 59 21.9 See also ...... 60 21.10References ...... 60 21.11Bibliography ...... 61 21.12External links ...... 61 CONTENTS v

22 Mitchell order 64 22.1 References ...... 64

23 65 23.1 Branches of model theory ...... 65 23.2 Universal algebra ...... 66 23.3 ...... 67 23.4 First-order logic ...... 67 23.5 Axiomatizability, elimination of quantifiers, and model-completeness ...... 68 23.6 Categoricity ...... 68 23.7 Model theory and set theory ...... 69 23.8 Other basic notions of model theory ...... 69 23.8.1 Reducts and expansions ...... 69 23.8.2 Interpretability ...... 70 23.8.3 Using the compactness and completeness ...... 70 23.8.4 Types ...... 70 23.9 History ...... 71 23.10See also ...... 71 23.11Notes ...... 71 23.12References ...... 71 23.12.1 Canonical textbooks ...... 71 23.12.2 Other textbooks ...... 71 23.12.3 Free online texts ...... 72

24 Normal measure 73 24.1 See also ...... 73 24.2 References ...... 73

25 74 25.1 Ordinals extend the natural ...... 75 25.2 Definitions ...... 77 25.2.1 Well-ordered sets ...... 77 25.2.2 Definition of an ordinal as an ...... 77 25.2.3 Von Neumann definition of ordinals ...... 77 25.2.4 Other definitions ...... 78 25.3 Transfinite ...... 78 25.4 Transfinite induction ...... 78 25.4.1 What is transfinite induction? ...... 78 25.4.2 Transfinite ...... 79 25.4.3 Successor and ordinals ...... 79 25.4.4 Indexing classes of ordinals ...... 79 25.4.5 Closed unbounded sets and classes ...... 80 vi CONTENTS

25.5 Arithmetic of ordinals ...... 80 25.6 Ordinals and cardinals ...... 80 25.6.1 Initial ordinal of a cardinal ...... 81 25.6.2 Cofinality ...... 81 25.7 Some “large” countable ordinals ...... 81 25.8 and ordinals ...... 82 25.9 Downward closed sets of ordinals ...... 82 25.10See also ...... 82 25.11Notes ...... 82 25.12References ...... 82 25.13External links ...... 83

26 84 26.1 Formal definition ...... 85 26.2 Examples ...... 85 26.3 Extrema ...... 85 26.4 Orders on the of partially ordered sets ...... 86 26.5 Sums of partially ordered sets ...... 86 26.6 Strict and non-strict partial orders ...... 87 26.7 Inverse and order ...... 87 26.8 Mappings between partially ordered sets ...... 87 26.9 Number of partial orders ...... 88 26.10Linear extension ...... 88 26.11In theory ...... 89 26.12Partial orders in topological spaces ...... 89 26.13Interval ...... 89 26.14See also ...... 89 26.15Notes ...... 90 26.16References ...... 90 26.17External links ...... 90

27 91 27.1 Example ...... 91 27.2 Properties ...... 92 27.3 Representing as functions ...... 92 27.4 Relation to binomial ...... 93 27.5 Algorithms ...... 93 27.6 Subsets of limited cardinality ...... 93 27.7 Power object ...... 94 27.8 Functors and quantifiers ...... 94 27.9 See also ...... 94 27.10Notes ...... 94 CONTENTS vii

27.11References ...... 95 27.12External links ...... 95

28 96 28.1 Formal definition ...... 96 28.2 Examples ...... 97 28.3 Uses ...... 98 28.4 Constructions ...... 98 28.5 Number of ...... 98 28.6 Interval ...... 99 28.7 See also ...... 99 28.8 References ...... 99

29 100 29.1 References ...... 100

30 Rank-into-rank 101 30.1 References ...... 101

31 103 31.1 Examples ...... 103 31.2 Properties ...... 103 31.3 See also ...... 104 31.4 References ...... 104

32 Scott’s trick 105 32.1 Application to ...... 105 32.2 References ...... 105

33 Sequence 106 33.1 Examples and notation ...... 107 33.1.1 Important examples ...... 107 33.1.2 Indexing ...... 108 33.1.3 Specifying a sequence by recursion ...... 109 33.2 Formal definition and basic properties ...... 109 33.2.1 Formal definition ...... 109 33.2.2 Finite and infinite ...... 110 33.2.3 Increasing and decreasing ...... 110 33.2.4 Bounded ...... 110 33.2.5 Other types of ...... 110 33.3 Limits and convergence ...... 111 33.3.1 Definition of convergence ...... 112 33.3.2 Applications and important results ...... 112 33.3.3 Cauchy sequences ...... 113 viii CONTENTS

33.4 ...... 113 33.5 Use in other fields of mathematics ...... 114 33.5.1 Topology ...... 114 33.5.2 Analysis ...... 114 33.5.3 Linear algebra ...... 115 33.5.4 Abstract algebra ...... 115 33.5.5 Set theory ...... 116 33.5.6 Computing ...... 116 33.5.7 Streams ...... 116 33.6 Types ...... 116 33.7 Related concepts ...... 117 33.8 Operations ...... 117 33.9 See also ...... 117 33.10References ...... 117 33.11External links ...... 118

34 Set theory 119 34.1 History ...... 120 34.2 Basic concepts and notation ...... 121 34.3 Some ontology ...... 122 34.4 Axiomatic set theory ...... 122 34.5 Applications ...... 123 34.6 Areas of study ...... 124 34.6.1 Combinatorial set theory ...... 124 34.6.2 Descriptive set theory ...... 124 34.6.3 theory ...... 124 34.6.4 Inner model theory ...... 124 34.6.5 Large cardinals ...... 125 34.6.6 Determinacy ...... 125 34.6.7 ...... 125 34.6.8 Cardinal invariants ...... 125 34.6.9 Set-theoretic topology ...... 126 34.7 Objections to set theory as a foundation for mathematics ...... 126 34.8 See also ...... 126 34.9 Notes ...... 126 34.10Further reading ...... 127 34.11External links ...... 127

35 Stanislaw Ulam 128 35.1 Poland ...... 128 35.2 Coming to America ...... 129 35.3 Manhattan Project ...... 130 CONTENTS ix

35.3.1 Hydrodynamical calculations of implosion ...... 131 35.3.2 of branching and multiplicative processes ...... 131 35.4 Post war Los Alamos ...... 131 35.4.1 Monte Carlo method ...... 132 35.4.2 Teller–Ulam design ...... 132 35.4.3 Fermi–Pasta–Ulam problem ...... 134 35.4.4 Nuclear propulsion ...... 135 35.5 Return to academia ...... 136 35.6 Impact and legacy ...... 138 35.7 Bibliography ...... 138 35.8 See also ...... 139 35.9 References ...... 140 35.10External links ...... 143

36 Stefan Banach 144 36.1 Life ...... 144 36.1.1 Early life ...... 144 36.1.2 Discovery by Steinhaus ...... 145 36.1.3 Interbellum ...... 145 36.1.4 World War II ...... 146 36.2 Contributions ...... 146 36.3 Quotes ...... 148 36.4 See also ...... 149 36.5 Notes ...... 149 36.6 References ...... 150 36.7 External links ...... 150

37 151 37.1 Definitions ...... 152 37.2 ⊂ and ⊃ symbols ...... 152 37.3 Examples ...... 152 37.4 Other properties of inclusion ...... 153 37.5 See also ...... 153 37.6 References ...... 153 37.7 External links ...... 154

38 155 38.1 Examples ...... 155 38.2 Properties ...... 155 38.3 Transitive closure ...... 155 38.4 Transitive models of set theory ...... 155 38.5 See also ...... 156 x CONTENTS

38.6 References ...... 156 38.7 External links ...... 156

39 Ultrafilter 157 39.1 Formal definition for ultrafilter on a set ...... 157 39.2 Completeness ...... 158 39.3 Generalization to partial orders ...... 158 39.4 Special case: ...... 158 39.5 Types and existence of ultrafilters ...... 158 39.6 Applications ...... 159 39.7 Ordering on ultrafilters ...... 160 39.8 Ultrafilters on ω ...... 160 39.9 See also ...... 160 39.10Notes ...... 160 39.11References ...... 161

40 Ultraproduct 162 40.1 Definition ...... 162 40.2 Examples ...... 163 40.3 Łoś's theorem ...... 163 40.3.1 Examples ...... 163 40.4 Ultralimit ...... 164 40.5 References ...... 164

41 165 41.1 Characterizations ...... 165 41.2 Properties ...... 165 41.3 Examples ...... 165 41.4 Without the axiom of choice ...... 166 41.5 See also ...... 166 41.6 References ...... 166 41.7 External links ...... 166

42 (mathematics) 167 42.1 In a specific context ...... 167 42.2 In ordinary mathematics ...... 167 42.3 In set theory ...... 168 42.4 In ...... 169 42.5 See also ...... 170 42.6 Notes ...... 170 42.7 References ...... 170 42.8 External links ...... 170 CONTENTS xi

43 Well-founded relation 171 43.1 Induction and recursion ...... 171 43.2 Examples ...... 172 43.3 Other properties ...... 172 43.4 Reflexivity ...... 173 43.5 References ...... 173

44 Zermelo–Fraenkel set theory 174 44.1 History ...... 174 44.2 ...... 175 44.2.1 1. Axiom of ...... 175 44.2.2 2. (also called the Axiom of foundation) ...... 175 44.2.3 3. of specification (also called the axiom schema of separation or of restricted comprehension) ...... 175 44.2.4 4. ...... 176 44.2.5 5. Axiom of ...... 176 44.2.6 6. Axiom schema of replacement ...... 177 44.2.7 7. Axiom of infinity ...... 178 44.2.8 8. ...... 178 44.2.9 9. Well-ordering theorem ...... 178 44.3 Motivation via the cumulative ...... 179 44.4 Metamathematics ...... 179 44.4.1 Independence ...... 180 44.5 Criticisms ...... 180 44.6 See also ...... 181 44.7 References ...... 181 44.8 External links ...... 182 44.9 Text and sources, contributors, and licenses ...... 183 44.9.1 Text ...... 183 44.9.2 Images ...... 188 44.9.3 Content license ...... 191 Chapter 1

Atom (measure theory)

In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller but positive measure. A measure which has no atoms is called non-atomic or atomless.

1.1 Definition

Given a measurable (X, Σ) and a measure µ on that space, a set A ⊂ X in Σ is called an atom if

µ(A) > 0 and for any measurable subset B ⊂ A with

µ(B) < µ(A) one has B has measure zero.

1.2 Examples

• Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra Σ be the power set of X. Define the measure µ of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons {i}, for i=1,2, ..., 9, 10 is an atom. • Consider the Lebesgue measure on the . This measure has no atoms.

1.3 Non-atomic measures

A measure which has no atoms is called non-atomic. In other words, a measure is non-atomic if for any measurable set A with µ(A) > 0 there exists a measurable subset B of A such that

µ(A) > µ(B) > 0.

A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set A with µ(A) > 0 one can construct a decreasing sequence of measurable sets

A = A1 ⊃ A2 ⊃ A3 ⊃ · · ·

1 2 CHAPTER 1. ATOM (MEASURE THEORY)

such that

µ(A) = µ(A1) > µ(A2) > µ(A3) > ··· > 0. This may not be true for measures having atoms; see the first example above. It turns out that non-atomic measures actually have a continuum of values. It can be proved that if μ is a non-atomic measure and A is a measurable set with µ(A) > 0, then for any b satisfying

µ(A) ≥ b ≥ 0 there exists a measurable subset B of A such that

µ(B) = b. This theorem is due to Wacław Sierpiński.[1][2] It is reminiscent of the intermediate value theorem for continuous functions. Sketch of proof of Sierpiński’s theorem on non-atomic measures. A slightly stronger statement, which however makes the proof easier, is that if (X, Σ, µ) is a non-atomic measure space and µ(X) = c , there exists a S : [0, c] → Σ that is monotone with respect to inclusion, and a right-inverse to µ :Σ → [0, c] . That is, there exists a one-parameter family of measurable sets S(t) such that for all 0 ≤ t ≤ t′ ≤ c

S(t) ⊂ S(t′), µ (S(t)) = t. The proof easily follows from Zorn’s lemma applied to the set of all monotone partial sections to µ :

Γ := {S : D → Σ: D ⊂ [0, c],S monotone, ∀t ∈ D (µ (S(t)) = t)}, ordered by inclusion of graphs, graph(S) ⊂ graph(S′). It’s then standard to show that every chain in Γ has an upper bound in Γ , and that any maximal of Γ has domain [0, c], proving the claim.

1.4 See also

• Atom (order theory) — an analogous concept in order theory • • , also known as an atomic event

1.5 Notes

[1] Sierpinski, W. (1922). “Sur les fonctions d'ensemble additives et continues” (PDF). Fundamenta Mathematicae (in French) 3: 240–246. [2] Fryszkowski, Andrzej (2005). Fixed Point Theory for Decomposable Sets (Topological Fixed Point Theory and Its Applica- tions). New York: Springer. p. 39. ISBN 1-4020-2498-3.

1.6 References

• Bruckner, Andrew M.; Bruckner, Judith B.; Thomson, Brian S. (1997). Real analysis. Upper Saddle River, N.J.: Prentice-Hall. p. 108. ISBN 0-13-458886-X. • Butnariu, Dan; Klement, E. P. (1993). Triangular -based measures and games with fuzzy coalitions. Dordrecht: Kluwer Academic. p. 87. ISBN 0-7923-2369-6. Chapter 2

Axiom of determinacy

The axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information. AD states that every such game in which both players choose natural numbers is determined; that is, one of the two players has a winning strategy. The axiom of determinacy is inconsistent with the axiom of choice (AC); the axiom of determinacy implies that all subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property. The last implies a weak form of the continuum hypothesis (namely, that every uncountable set of reals has the same cardinality as the full set of reals). Furthermore, AD implies the consistency of Zermelo–Fraenkel set theory (ZF). Hence, as a consequence of the incompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF.

2.1 Types of game that are determined

Not all games require the axiom of determinacy to prove them determined. Games whose winning sets are closed are determined. These correspond to many naturally defined infinite games. It was shown in 1975 by Donald A. Martin that games whose winning set is a Borel set are determined. It follows from the existence of sufficient large cardinals that all games with winning set a projective set are determined (see Projective determinacy), and that AD holds in L(R).

2.2 Incompatibility of the axiom of determinacy with the axiom of choice

The set S1 of all first player strategies in an ω-game G has the same cardinality as the continuum. The same is true of the set S2 of all second player strategies. We note that the cardinality of the set SG of all sequences possible in G is also the continuum. Let A be the subset of SG of all sequences which make the first player win. With the axiom of choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portion does not have the cardinality of the continuum. We create a counterexample by transfinite induction on the set of strategies under this well ordering: We start with the set A undefined. Let T be the “time” whose axis has length continuum. We need to consider all strategies {s1(T)} of the first player and all strategies {s2(T)} of the second player to make sure that for every strategy there is a strategy of the other player that wins against it. For every strategy of the player considered we will generate a sequence which gives the other player a win. Let t be the time whose axis has length ℵ0 and which is used during each game sequence.

1. Consider the strategy {s1(T)} of the first player. 2. Go through the entire game, generating (together with the first player’s strategy s1(T)) a sequence {a(1), b(2), a(3), b(4),...,a(t), b(t+1),...}. 3. Decide that this sequence does not belong to A, i.e. s1(T) lost.

3 4 CHAPTER 2. AXIOM OF DETERMINACY

4. Consider the strategy {s2(T)} of the second player.

5. Go through the next entire game, generating (together with the second player’s strategy s2(T)) a sequence {c(1), d(2), c(3), d(4),...,c(t), d(t+1),...}, making sure that this sequence is different from {a(1), b(2), a(3), b(4),...,a(t), b(t+1),...}.

6. Decide that this sequence belongs to A, i.e. s2(T) lost.

7. Keep repeating with further strategies if there are any, making sure that sequences already considered do not become generated again. (We start from the set of all sequences and each time we generate a sequence and refute a strategy we project the generated sequence onto first player moves and onto second player moves, and we take away the two resulting sequences from our set of sequences.)

8. For all sequences that did not come up in the above consideration arbitrarily decide whether they belong to A, or to the of A.

Once this has been done we have a game G. If you give me a strategy s1 then we considered that strategy at some time T = T(s1). At time T, we decided an of s1 that would be a loss of s1. Hence this strategy fails. But this is true for an arbitrary strategy; hence the axiom of determinacy and the axiom of choice are incompatible.

2.3 Infinite logic and the axiom of determinacy

Many different versions of infinitary logic were proposed in the late 20th century. One reason that has been given for believing in the axiom of determinacy is that it can be written as follows (in a version of infinite logic): ∀G ⊆ Seq(S): ∀a ∈ S : ∃a′ ∈ S : ∀b ∈ S : ∃b′ ∈ S : ∀c ∈ S : ∃c′ ∈ S... :(a, a′, b, b′, c, c′...) ∈ G OR ∃a ∈ S : ∀a′ ∈ S : ∃b ∈ S : ∀b′ ∈ S : ∃c ∈ S : ∀c′ ∈ S... :(a, a′, b, b′, c, c′...)/∈ G Note: Seq(S) is the set of all ω -sequences of S. The sentences here are infinitely long with a countably infinite list of quantifiers where the appear. In an infinitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantifiers that are true for finite formulas, such as ∀a : ∃b : ∀c : ∃d : R(a, b, c, d) OR ∃a : ∀b : ∃c : ∀d : ¬R(a, b, c, d) .

2.4 Large cardinals and the axiom of determinacy

The consistency of the axiom of determinacy is closely related to the question of the consistency of axioms. By a theorem of Woodin, the consistency of Zermelo–Fraenkel set theory without choice (ZF) together with the axiom of determinacy is equivalent to the consistency of Zermelo–Fraenkel set theory with choice (ZFC) together with the existence of infinitely many Woodin cardinals. Since Woodin cardinals are strongly inaccessible, if AD is consistent, then so are an infinity of inaccessible cardinals. Moreover, if to the hypothesis of an infinite set of Woodin cardinals is added the existence of a measurable cardinal larger than all of them, a very strong theory of Lebesgue measurable sets of reals emerges, as it is then provable that the axiom of determinacy is true in L(R), and therefore that every set of real numbers in L(R) is determined.

2.5 See also

• Axiom of real determinacy (ADR)

• AD+, a variant of the axiom of determinacy formulated by Woodin

• Axiom of quasi-determinacy (ADQ)

• Martin measure 2.6. REFERENCES 5

2.6 References

• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540- 44085-2. • Kanamori, Akihiro (2000). The Higher Infinite (2nd ed.). Springer. ISBN 3-540-00384-3.

• Martin, Donald A.; Steel, John R. (Jan 1989). “A Proof of Projective Determinacy”. Journal of the American Mathematical Society 2 (1): 71–125. doi:10.2307/1990913. JSTOR 1990913. • Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

• Mycielski, Jan; Steinhaus, H. (1962). “A mathematical axiom contradicting the axiom of choice”. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 10: 1–3. ISSN 0001-4117. MR 0140430. • Woodin, W. Hugh (1988). “Supercompact cardinals, sets of reals, and weakly homogeneous trees”. Proceedings of the National Academy of Sciences of the United States of America 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587. PMC 282022. PMID 16593979.

2.7 Further reading

• Philipp Rohde, On Extensions of the Axiom of Determinacy, Thesis, Department of Mathematics, University of Bonn, Germany, 2001

• Telgársky, R.J. Topological Games: On the 50th Anniversary of the Banach-Mazur Game, Rocky Mountain J. Math. 17 (1987), pp. 227–276. (3.19 MB) Chapter 3

Cardinal number

This article is about the mathematical concept. For number words indicating quantity (“three” apples, “four” birds, etc.), see Cardinal number (). In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a : the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence () between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets. There is a transfinite sequence of cardinal numbers:

0, 1, 2, 3, . . . , n, . . . ; ℵ0, ℵ1, ℵ2,..., ℵα,....

This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and . In category theory, the cardinal numbers form a skeleton of the .

3.1 History

The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874– 1884. Cardinality can be used to compare an aspect of finite sets; e.g. the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three (this is established by the existence of a bijection, i.e. a one-to-one correspondence, between the two sets; e.g. {1->4, 2->5, 3->6}). Cantor applied his concept of bijection to infinite sets;[1] e.g. the set of natural numbers N = {0, 1, 2, 3, ...}. Thus, all sets having a bijection with N he called denumerable (countably infinite) sets and they all have the same cardinal number. This cardinal number is called ℵ0 , aleph-null. He called the cardinal numbers of these infinite sets, transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N, even though this might appear to run contrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable (which implies that the set of all rational numbers is denumerable), and later proved that the set of all algebraic numbers is also denumerable. Each z may be encoded as a finite sequence of which are the coefficients

6 3.1. HISTORY 7 X Y 1 D

2 B 3 C 4 A

A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.

in the polynomial equation of which it is the solution, i.e. the ordered n- (a0, a1, ..., an), ai ∈ Z together with a pair of rationals (b0, b1) such that z is the unique root of the polynomial with coefficients (a0, a1, ..., an) that lies in the interval (b0, b1). In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbers has cardinality greater than that of N. His original presentation used a complex with nested intervals, but in an 1891 paper he proved the same result using his ingenious but simple diagonal argument. The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the c for it. Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ( ℵ0 , aleph-null) and that for every cardinal number, there is a next-larger cardinal

(ℵ1, ℵ2, ℵ3, ··· ).

His continuum hypothesis is the that c is the same as ℵ1 . This hypothesis has been found to be inde- pendent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standard assumptions. 8 CHAPTER 3. CARDINAL NUMBER

Aleph null, the smallest infinite cardinal

3.2 Motivation

In informal use, a cardinal number is what is normally referred to as a counting number, provided that 0 is included: 0, 1, 2, .... They may be identified with the natural numbers beginning with 0. The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic. More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set which has exactly the right size, e.g. 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c} which has 3 elements. However when dealing with infinite sets it is essential to distinguish between the two — the two notions are in fact different for infinite sets. Considering the position aspect leads to ordinal numbers, while the size aspect is generalized by the cardinal numbers described here. The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or “bigness” of a set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions. 3.3. FORMAL DEFINITION 9

A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y. An injective mapping identifies each element of the set X with a unique element of the set Y. This is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size we would observe that there is a mapping:

1 → a 2 → b 3 → c

which is injective, and hence conclude that Y has cardinality greater than or equal to X. Note the element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily an injective and onto mapping. The advantage of this notion is that it can be extended to infinite sets. We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if there exists a bijection between X and Y. By the Schroeder–Bernstein theorem, this is equivalent to there being both an injective mapping from X to Y and an injective mapping from Y to X. We then write |X| = |Y|. The cardinal number of X itself is often defined as the least ordinal a with |a| = |X|. This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects. The classic example used is that of the infinite hotel paradox, also called Hilbert’s paradox of the Grand Hotel. Suppose you are an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:

1 ↔ 2 2 ↔ 3 3 ↔ 4 ... n ↔ n + 1 ...

In this way we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...} since a bijection between the first and the second has been shown. This motivates the definition of an infinite set being any set which has a proper subset of the same cardinality; in this case {2,3,4,...} is a proper subset of {1,2,3,...}. When considering these large objects, we might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it doesn't; by considering the above example we can see that if some object “one greater than infinity” exists, then it must have the same cardinality as the infinite set we started out with. It is possible to use a different formal notion for number, called ordinals, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers. It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. This can be visualized using Cantor’s diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite car- dinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals. Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to as equipotence, equipollence, or . It is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.

3.3 Formal definition

Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal α such that there is a bijection between X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not 10 CHAPTER 3. CARDINAL NUMBER

assumed we need to do something different. The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and ) is as the class [X] of all sets that are equinumerous with X. This does not work in ZFC or other related systems of axiomatic set theory because if X is non-empty, this collection is too large to be a set. In fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × X and so by the axiom of limitation of size, [X] is a proper class. The definition does work however in and in and related systems. However, if we restrict from this class to those equinumerous with X that have the least rank, then it will work (this is a trick due to Dana Scott:[2] it works because the collection of objects with any given rank is a set). Formally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists an injective function from X to Y. The Cantor–Bernstein–Schroeder theorem states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|. The axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y| or |Y| ≤ |X|.[3][4] A set X is Dedekind-infinite if there exists a proper subset Y of X with |X| = |Y|, and Dedekind-finite if such a subset doesn't exist. The finite cardinals are just the natural numbers, i.e., a set X is finite if and only if |X| = |n| = n for some natural number n. Any other set is infinite. Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also be proved that the cardinal ℵ0 (aleph null or aleph-0, where aleph is the first letter in the , represented ℵ ) of the set of natural numbers is the smallest infinite cardinal, i.e. that any infinite set has a subset of cardinality ℵ0. The next larger cardinal is denoted by ℵ1 and so on. For every ordinal α there is a cardinal number ℵα, and this list exhausts all infinite cardinal numbers.

3.4 Cardinal arithmetic

We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.

3.4.1 Successor cardinal

For more details on this topic, see Successor cardinal.

If the axiom of choice holds, every cardinal κ has a successor κ+ > κ, and there are no cardinals between κ and its successor. For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal.

3.4.2 Cardinal addition

If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality, e.g., replace X by X×{0} and Y by Y×{1}.

|X| + |Y | = |X ∪ Y |.

Zero is an additive identity κ + 0 = 0 + κ = κ. Addition is associative (κ + μ) + ν = κ + (μ + ν). Addition is commutative κ + μ = μ + κ. Addition is non-decreasing in both arguments:

(κ ≤ µ) → ((κ + ν ≤ µ + ν) and (ν + κ ≤ ν + µ)).

Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either κ or μ is infinite, then

κ + µ = max{κ, µ} . 3.4. CARDINAL ARITHMETIC 11

Subtraction

Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.

3.4.3 Cardinal multiplication

The product of cardinals comes from the cartesian product.

|X| · |Y | = |X × Y |

κ·0 = 0·κ = 0. κ·μ = 0 → (κ = 0 or μ = 0). One is a multiplicative identity κ·1 = 1·κ = κ. Multiplication is associative (κ·μ)·ν = κ·(μ·ν). Multiplication is commutative κ·μ = μ·κ. Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ). Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ. Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then

κ · µ = max{κ, µ}.

Division

Assuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π.

3.4.4 Cardinal exponentiation

Exponentiation is given by

|X||Y | = XY

where XY is the set of all functions from Y to X.

κ0 = 1 (in particular 00 = 1), see empty function. If 1 ≤ μ, then 0μ = 0. 1μ = 1. κ1 = κ. κμ + ν = κμ·κν. κμ · ν = (κμ)ν. (κ·μ)ν = κν·μν.

Exponentiation is non-decreasing in both arguments:

(1 ≤ ν and κ ≤ μ) → (νκ ≤ νμ) and (κ ≤ μ) → (κν ≤ μν). 12 CHAPTER 3. CARDINAL NUMBER

Note that 2|X| is the cardinality of the power set of the set X and Cantor’s diagonal argument shows that 2|X| > |X| for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2κ). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.) All the remaining in this section assume the axiom of choice:

If κ and μ are both finite and greater than 1, and ν is infinite, then κν = μν. If κ is infinite and μ is finite and non-zero, then κμ = κ.

If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:

Max (κ, 2μ) ≤ κμ ≤ Max (2κ, 2μ).

Using König’s theorem, one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ.

Roots

Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying νµ = κ will be κ.

Logarithms

Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying µλ = κ . However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy νλ = κ . The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of possess.[5][6][7]

3.5 The continuum hypothesis

ℵ0 The continuum hypothesis (CH) states that there are no cardinals strictly between ℵ0 and 2 . The latter cardinal number is also often denoted by c ; it is the cardinality of the continuum (the set of real numbers). In this case ℵ0 2 = ℵ1. The generalized continuum hypothesis (GCH) states that for every infinite set X, there are no cardinals strictly between | X | and 2| X |. The continuum hypothesis is independent of the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice (ZFC).

3.6 See also

3.7 References

Notes

[1] Dauben 1990, pg. 54

[2] Deiser, Oliver (May 2010). “On the Development of the Notion of a Cardinal Number”. History and Philosophy of Logic 31 (2): 123–143. doi:10.1080/01445340903545904.

[3] Enderton, Herbert. “Elements of Set Theory”, Academic Press Inc., 1977. ISBN 0-12-238440-7

[4] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "Über das Problem der Wohlordnung”, Math. Ann (Leipzig: B. G. Teubner), Bd. 76 (4): 438–443, ISSN 0025-5831 3.8. EXTERNAL LINKS 13

[5] Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathe- matics 1315, Springer-Verlag.

[6] Eduard Čech, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.

[7] D.A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.

Bibliography

• Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton: Princeton University Press, ISBN 0691-02447-2

• Hahn, Hans, Infinity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. New York: Simon and Schuster, 1956.

• Halmos, Paul, . Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer- Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

3.8 External links

• Hazewinkel, Michiel, ed. (2001), “Cardinal number”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4 • Weisstein, Eric W., “Cardinal Number”, MathWorld.

• Cardinality at ProvenMath proofs of the basic theorems on cardinality. Chapter 4

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathe- matical objects) that can be unambiguously defined by a property that all its members share. The precise definition of “class” depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as Von Neumann–Bernays–Gödel set theory, axiomatize the notion of “proper class”, e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. Outside set theory, the word “class” is sometimes used synonymously with “set”. This usage dates from a histor- ical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of “classes” in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.

4.1 Examples

The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of forms a proper class) is called a large category. The surreal numbers are a proper class of objects that have the properties of a field. Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete .

4.2 Paradoxes

The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that “all classes are sets”. With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper. For example, Russell’s paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali- Forti paradox suggests that the class of all ordinal numbers is proper.

4.3 Classes in formal set theories

ZF set theory does not formalize the notion of classes, so each formula with classes must be reduced syntactically to a formula without classes.[1] For example, one can reduce the formula A = {x | x = x} to ∀x.(x ∈ A ↔ x = x) . Semantically, in a metalanguage, the classes can be described as equivalence classes of logical formulas: If A

14 4.4. REFERENCES 15 is a structure interpreting ZF, then the object language class builder expression {x | ϕ} is interpreted in A by the collection of all the elements from the domain of A on which λx.ϕ holds; thus, the class can be described as the set of all predicates equivalent to ϕ (including ϕ itself). In particular, one can identify the “class of all sets” with the set of all predicates equivalent to x=x. Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as “classes”. In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula Φ(x, y) with the property that for any set x there is no more than one set y such that the pair (x,y) satisfies Φ . For example, the class function mapping each set to its successor may be expressed as the formula y = x ∪ {x} . The fact that the (x,y) satisfies Φ may be expressed with the shorthand notation Φ(x) = y . Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a conservative extension of ZF. Morse–Kelley set theory admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its class existence axioms. This causes MK to be strictly stronger than both NBG and ZF. In other set theories, such as New Foundations or the theory of semisets, the concept of “proper class” still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a has proper classes which are subclasses of sets.

4.4 References

[1] http://us.metamath.org/mpegif/abeq2.html

• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7 • Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag

4.5 External links

• Weisstein, Eric W., “Set Class”, MathWorld. Chapter 5

Club filter

In mathematics, particularly in set theory, if κ is a regular uncountable cardinal then club(κ) , the filter of all sets containing a club subset of κ , is a κ -complete filter closed under diagonal intersection called the club filter. To see that this is a filter, note that κ ∈ club(κ) since it is thus both closed and unbounded (see club set). If x ∈ club(κ) then any subset of κ containing x is also in club(κ) , since x , and therefore anything containing it, contains a club set. ⟨ ⟩ It is a κ -complete filter because the intersection of fewer than∩ κ club sets is a club set. To see this, suppose Ci i<α is a sequence of club sets where α < κ . Obviously C = Ci is closed, since any sequence which appears in C appears in every Ci , and therefore its limit is also in every Ci . To show that it is unbounded, take some β < κ . Let ⟨β1,i⟩ be an increasing sequence with β1,1 > β and β1,i ∈ Ci for every i < α . Such a sequence can be constructed, since every Ci is unbounded. Since α < κ and κ is regular, the limit of this sequence is less than κ . We call it β2 , and define a new sequence ⟨β2,i⟩ similar to the previous sequence. We can repeat this process, getting a sequence of sequences ⟨βj,i⟩ where each element of a sequence is greater than every member of the previous sequences. Then for each i < α , ⟨βj,i⟩ is an increasing sequence contained in Ci , and all these sequences have the same limit (the limit of ⟨βj,i⟩ ). This limit is then contained in every Ci , and therefore C , and is greater than β . ⟨ ⟩ To see that club(κ) is closed under diagonal intersection,∪ let Ci , i < κ be a sequence of club sets, and let C = ∆i<κCi . To show C is closed, suppose S ⊆ α < κ and S = α . Then for each γ ∈ S , γ ∈ Cβ for all β < γ ∈ ∈ . Since each Cβ is closed, α Cβ for all β < α , so α C . To show C is∩ unbounded, let α < κ , and define a sequence ξi , i < ω as follows: ξ0 = α , and ξi+1 is the minimal element of γ<ξ Cγ ∪such that ξi+1 > ξi . Such i ∈ an element exists since by the above, the intersection of ξi club sets is club. Then ξ = i<ω ξi > α and ξ C , since it is in each Ci with i < ξ .

5.1 References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.

This article incorporates material from club filter on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

16 Chapter 6

Club set

In mathematics, particularly in and set theory, a club set is a subset of a which is closed under the , and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of “closed and unbounded”.

6.1 Formal definition

Formally, if κ is a limit ordinal, then a set C ⊆ κ is closed in κ if and only if for every α < κ , if sup(C ∩α) = α ≠ 0 , then α ∈ C . Thus, if the limit of some sequence from C is less than κ , then the limit is also in C . If κ is a limit ordinal and C ⊆ κ then C is unbounded in κ if for any α < κ , there is some β ∈ C such that α < β . If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals). For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded. The set of all limit ordinals α < κ is closed unbounded in κ ( κ regular). In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous). More generally, if X is a nonempty set and λ is a cardinal, then C ⊆ [X]λ is club if every union of a subset of C is in C and every subset of X of cardinality less than λ is contained in some element of C (see ).

6.2 The closed unbounded filter

⟨ ⟩ Let κ be a limit ordinal of uncountable∩ cofinality λ . For some α < λ , let Cξ : ξ < α be a sequence of closed unbounded subsets of κ . Then ξ<α Cξ is also closed unbounded. To see this, one can note that an intersection of closed sets is always closed, so we just need to show that this intersection is unbounded. So fix any β0 < κ , and for ξ each n<ω choose from each Cξ an element βn+1 > βn , which is possible because each is unbounded. Since this is a collection of fewer than λ ordinals, all less than κ , their least upper bound must also be less than κ , so we can call it βn+1 . This process generates a countable sequence β0, β1, β2,.... The limit of this sequence must in fact also ξ ξ ξ be the limit of the sequence β0 , β1 , β2 ,..., and since each Cξ is closed and λ is uncountable, this limit must be in each Cξ , and therefore this limit is an element of the intersection that is above β0 , which shows that the intersection is unbounded. QED. From this, it can be seen that if κ is a regular cardinal, then {S ⊂ κ : ∃C ⊂ S that such C in unbounded closed is κ} is a non-principal κ -complete filter on κ . If κ is a regular cardinal then club sets are also closed under diagonal intersection. In fact, if κ is regular and F is any filter on κ , closed under diagonal intersection, containing all sets of the form {ξ < κ : ξ ≥ α} for α < κ , then F must include all club sets.

17 18 CHAPTER 6. CLUB SET

6.3 See also

• Club filter

• Stationary set • Clubsuit

6.4 References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2. • Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. ISBN 0-486-42079-5 • This article incorporates material from Club on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Chapter 7

Continuum hypothesis

This article is about the hypothesis in set theory. For the assumption in fluid mechanics, see Fluid mechanics.

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert’s 23 problems presented in the year 1900. Τhe answer to this problem is independent of ZFC set theory, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by , complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term the continuum for the real numbers. It is abbreviated CH.

7.1 Cardinality of infinite sets

Main article: Cardinal number

Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspon- dence) between them. Intuitively, for two sets S and T to have the same cardinality means that it is possible to “pair off” elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green} . With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers, and more real numbers than rational numbers. However, this intuitive analysis does not take account of the fact that all three sets are infinite. It turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set of integers: they are both countable sets. Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (see Cantor’s first uncountability proof and Cantor’s diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. Equivalently, as the cardinality of the integers is ℵ0 ("aleph-naught") and the cardinality of the ℵ real numbers is 2 0 (i.e. it equals the cardinality of the power set of the integers), the continuum hypothesis says that there is no set S for which

ℵ0 ℵ0 < |S| < 2 .

19 20 CHAPTER 7. CONTINUUM HYPOTHESIS

Assuming the axiom of choice, there is a smallest cardinal number ℵ1 greater than ℵ0 , and the continuum hypothesis is in turn equivalent to the equality

ℵ0 2 = ℵ1. A consequence of the continuum hypothesis is that every infinite subset of the real numbers either has the same cardinality as the integers or the same cardinality as the entire set of the reals. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH) which says that for all ordinals α

ℵα 2 = ℵα+1. That is, GCH asserts that the cardinality of the power set of any infinite set is the smallest cardinality greater than that of the set.

7.2 Independence from ZFC

Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain (Dauben 1990). It became the first on David Hilbert’s list of important open questions that was presented at the International Congress of Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that point not yet formulated. Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo–Fraenkel set theory (ZF), even if the axiom of choice is adopted (ZFC) (Gödel (1940)). Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either (Cohen (1963)& Cohen (1964)). Hence, CH is independent of ZFC. Both of these results assume that the Zermelo–Fraenkel axioms are consistent; this assumption is widely believed to be true. Cohen was awarded the Fields Medal in 1966 for his proof. The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory. As a result of its independence, many substantial conjectures in those fields have subsequently been shown to be independent as well. So far, CH appears to be independent of all known large cardinal axioms in the context of ZFC. The independence from ZFC means that proving or disproving the CH within ZFC is impossible. Gödel and Cohen’s negative results are not universally accepted as disposing of the hypothesis. Hilbert’s problem remains an active topic of research; see Woodin (2001) and Koellner (2011a) for an overview of the current research status. The continuum hypothesis was not the first statement shown to be independent of ZFC. An immediate consequence of Gödel’s incompleteness theorem, which was published in 1931, is that there is a formal statement (one for each appropriate Gödel numbering scheme) expressing the consistency of ZFC that is independent of ZFC, assuming that ZFC is consistent. The continuum hypothesis and the axiom of choice were among the first mathematical statements shown to be independent of ZF set theory. These proofs of independence were not completed until Paul Cohen developed forcing in the 1960s. They all rely on the assumption that ZF is consistent. These proofs are called proofs of relative consistency (see Forcing (mathematics)).

7.3 Arguments for and against CH

Gödel believed that CH is false and that his proof that CH is consistent with ZFC only shows that the Zermelo– Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a platonist and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a formalist (Goodman 1979), also tended towards rejecting CH. Historically, mathematicians who favored a “rich” and “large” universe of sets were against CH, while those favoring a “neat” and “controllable” universe favored CH. Parallel arguments were made for and against the axiom of con- structibility, which implies CH. More recently, has pointed out that ontological maximalism can actually be used to argue in favor of CH, because among models that have the same reals, models with “more” sets of reals have a better chance of satisfying CH (Maddy 1988, p. 500). 7.4. THE GENERALIZED CONTINUUM HYPOTHESIS 21

Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 by Skolem, even before Gödel’s first incompleteness theorem. Skolem argued on the basis of what is now known as Skolem’s paradox, and it was later supported by the independence of CH from the axioms of ZFC, since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false (Kunen 1980, p. 171). At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling presented an argument against CH by showing that the negation of CH is equivalent to Freiling’s axiom of symmetry, a statement about . Freiling this axiom is “intuitively true” but others have disagreed. A difficult argument against CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000 (Woodin 2001a, 2001b). Foreman (2003) does not reject Woodin’s argument outright but urges caution. Solomon Feferman (2011) has made a complex philosophical argument that CH is not a definite mathematical prob- lem. He proposes a theory of “definiteness” using a semi-intuitionistic subsystem of ZF that accepts for bounded quantifiers but uses intuitionistic logic for unbounded ones, and suggests that a proposition ϕ is mathemati- cally “definite” if the semi-intuitionistic theory can prove (ϕ ∨ ¬ϕ) . He conjectures that CH is not definite according to this notion, and proposes that CH should therefore be considered not to have a . Peter Koellner (2011b) wrote a critical commentary on Feferman’s article. Joel David Hamkins proposes a multiverse approach to set theory and argues that “the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.” (Hamkins 2012). In a related vein, wrote that he does “not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to ZFC.” (Shelah 2003).

7.4 The generalized continuum hypothesis

The generalized continuum hypothesis (GCH) states that if an infinite set’s cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S. That is, for any infinite cardinal λ there is no cardinal κ such that λ < κ < 2λ. GCH is equivalent to:

ℵα ℵα+1 = 2 for every ordinal α. (occasionally called Cantor’s aleph hypothesis)

The beth numbers provide an alternate notation for this condition: ℵα = ℶα for every ordinal α. This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers. It was first suggested by Jourdain (1905). Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some Aleph number, and thus can ℵ be ordered. This is done by showing that n is smaller than 2 0+n which is smaller than its own Hartogs number — ℵ ℵ this uses the equality 2 0+n = 2 · 2 0+n ; for the full proof, see Gillman (2002). Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen’s model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed by Cohen to prove Easton’s theorem, which shows it is consistent with ZFC for arbitrarily large cardinals ℵα to ℵα fail to satisfy 2 = ℵα+1. Much later, Foreman and Woodin proved that (assuming the consistency of very large cardinals) it is consistent that 2κ > κ+ holds for every infinite cardinal κ. Later Woodin extended this by showing the consistency of 2κ = κ++ for every κ . A recent result of Carmi Merimovich shows that, for each n≥1, it is consistent with ZFC that for each κ, 2κ is the nth successor of κ. On the other hand, László Patai (1930) proved, that if γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite. For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsets of B. Thus for any infinite cardinals A and B, 22 CHAPTER 7. CONTINUUM HYPOTHESIS

A < B → 2A ≤ 2B.

If A and B are finite, the stronger inequality

A < B → 2A < 2B holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.

7.4.1 Implications of GCH for cardinal exponentiation

Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, one ℵβ can deduce from it the values of cardinal exponentiation in all cases. It implies that ℵα is (see: Hayden & Kennison (1968), page 147, exercise 76):

ℵβ+1 when α ≤ β+1;

ℵα when β+1 < α and ℵβ < cf(ℵα) where cf is the cofinality operation; and

ℵα+1 when β+1 < α and ℵβ ≥ cf(ℵα) .

7.5 See also

• Aleph number

•

• Cardinality

• Ω-logic

• Wetzel’s problem

7.6 References

• Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: Dover Publications. ISBN 978-0-486-46921-8.

• Cohen, Paul J. (December 15, 1963). “The Independence of the Continuum Hypothesis”. Proceedings of the National Academy of Sciences of the United States of America 50 (6): 1143–1148. doi:10.1073/pnas.50.6.1143. JSTOR 71858. PMC 221287. PMID 16578557.

• Cohen, Paul J. (January 15, 1964). “The Independence of the Continuum Hypothesis, II”. Proceedings of the National Academy of Sciences of the United States of America 51 (1): 105–110. doi:10.1073/pnas.51.1.105. JSTOR 72252. PMC 300611. PMID 16591132.

• Dales, H. G.; Woodin, W. H. (1987). An Introduction to Independence for Analysts. Cambridge.

• Dauben, Joseph Warren (1990). Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press. pp. 134–137. ISBN 9780691024479.

• Enderton, Herbert (1977). Elements of Set Theory. Academic Press.

• Feferman, Solomon (2011). “Is the Continuum Hypothesis a definite mathematical problem?" (PDF). Exploring the Frontiers of Independence (Harvard lecture series).

• Foreman, Matt (2003). “Has the Continuum Hypothesis been Settled?" (PDF). Retrieved February 25, 2006. 7.7. EXTERNAL LINKS 23

• Freiling, Chris (1986). “Axioms of Symmetry: Throwing Darts at the Real Number Line”. Journal of Symbolic Logic (Association for Symbolic Logic) 51 (1): 190–200. doi:10.2307/2273955. JSTOR 2273955. • Gödel, K. (1940). The Consistency of the Continuum-Hypothesis. Princeton University Press. • Gillman, Leonard (2002). “Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis” (PDF). American Mathematical Monthly 109. • Gödel, K.: What is Cantor’s Continuum Problem?, reprinted in Benacerraf and Putnam’s collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel’s arguments against CH. • Goodman, Nicolas D. (1979). “Mathematics as an objective science”. The American Mathematical Monthly 86 (7): 540–551. doi:10.2307/2320581. MR 542765. This view is often called formalism. Positions more or less like this may be found in Haskell Curry [5], Abraham Robinson [17], and Paul Cohen [4]. • Joel David Hamkins. The set-theoretic multiverse. Rev. Symb. Log. 5 (2012), no. 3, 416–449. • Seymour Hayden and John F. Kennison: Zermelo–Fraenkel Set Theory (1968), Charles E. Merrill Publishing Company, Columbus, Ohio. • Jourdain, Philip E. B. (1905). “On transfinite cardinal numbers of the exponential form”. Philosophical Mag- azine, Series 6 9: 42–56. doi:10.1080/14786440509463254. • Koellner, Peter (2011a). “The Continuum Hypothesis” (PDF). Exploring the Frontiers of Independence (Har- vard lecture series). • Koellner, Peter (2011b). “Feferman On the Indefiniteness of CH” (PDF). • Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam: North-Holland. ISBN 978-0-444-85401-8. • Maddy, Penelope (June 1988). “Believing the Axioms, I”. Journal of Symbolic Logic (Association for Symbolic Logic) 53 (2): 481–511. doi:10.2307/2274520. JSTOR 2274520. • Martin, D. (1976). “Hilbert’s first problem: the continuum hypothesis,” in Mathematical Developments Arising from Hilbert’s Problems, Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1 • McGough, Nancy. “The Continuum Hypothesis”. • Merimovich, Carmi (2007). “A power function with a fixed finite gap everywhere”. Journal of Symbolic Logic 72 (2): 361–417. doi:10.2178/jsl/1185803615. • Moore, Gregory H. (2011). “Early history of the generalized continuum hypothesis: 1878–1938”. Bull. Sym- bolic Logic 17 (4): 489–532. doi:10.2178/bsl/1318855631. MR 2896574. • Shelah, Saharon (2003). “Logical dreams”. Bull. Amer. Math. Soc. (N.S.) 40 (2): 203–228. doi:10.1090/s0273- 0979-03-00981-9. • Woodin, W. Hugh (2001a). “The Continuum Hypothesis, Part I” (PDF). Notices of the AMS 48 (6): 567–576. • Woodin, W. Hugh (2001b). “The Continuum Hypothesis, Part II” (PDF). Notices of the AMS 48 (7): 681–690.

German literature

• Cantor, Georg (1878). “Ein Beitrag zur Mannigfaltigkeitslehre”. Journal für die Reine und Angewandte Math- ematik 84: 242–258. doi:10.1515/crll.1878.84.242. reihe”. Mathematische und naturwissenschaftliche Berichte aus-א Patai, L. (1930). “Untersuchungen über die • Ungarn 37: 127–142.

7.7 External links

• Szudzik, Matthew and Weisstein, Eric W., “Continuum Hypothesis”, MathWorld.

This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Chapter 8

Critical point (set theory)

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself.[1] Suppose that j : N → M is an elementary embedding where N and M are transitive classes and j is definable in N by a formula of set theory with parameters from N. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(ω)=ω. If j(α)=α for all α<κ and j(κ)>κ, then κ is said to be the critical point of j. If N is V, then κ (the critical point of j) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a <κ-complete, non-principal ultrafilter over κ. Specifically, one may take the filter to be {A|A ⊆ κ∧κ ∈ j(A)} . Generally, there will be many other <κ-complete, non-principal ultrafilters over κ. However, j might be different from the ultrapower(s) arising from such filter(s). If N and M are the same and j is the identity function on N, then j is called “trivial”. If transitive class N is an inner model of ZFC and j has no critical point, i.e. every ordinal maps to itself, then j is trivial.

8.1 References

[1] Jech, Thomas (2002). Set Theory. Berlin: Springer-Verlag. ISBN 3-540-44085-2. p. 323

24 Chapter 9

Dana Scott

Dana Stewart Scott (born October 11, 1932) is the emeritus Hillman University Professor of , Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. His research career involved computer science, mathematics, and philosophy. His work on automata theory earned him the ACM Turing Award in 1976, while his collaborative work with Christopher Strachey in the 1970s laid the foundations of modern approaches to the semantics of programming languages. He has worked also on modal logic, topology, and category theory.

9.1 Early career

He received his BA in Mathematics from the University of California, Berkeley, in 1954. He wrote his Ph.D. thesis on Convergent Sequences of Complete Theories under the supervision of Alonzo Church while at Princeton, and defended his thesis in 1958. Solomon Feferman (2005) writes of this period:

Scott began his studies in logic at Berkeley in the early 50s while still an undergraduate. His unusual abilities were soon recognized and he quickly moved on to graduate classes and seminars with Tarski and became part of the group that surrounded him, including me and Richard Montague; so it was at that time that we became friends. Scott was clearly in line to do a Ph. D. with Tarski, but they had a falling out for reasons explained in our biography.[2] Upset by that, Scott left for Princeton where he finished with a Ph. D. under Alonzo Church. But it was not long before the relationship between them was mended to the point that Tarski could say to him, “I hope I can call you my student.”

After completing his Ph.D. studies, he moved to the University of Chicago, working as an instructor there until 1960. In 1959, he published a joint paper with Michael O. Rabin, a colleague from Princeton, entitled Finite Automata and Their ,[3] which introduced the idea of nondeterministic machines to automata theory. This work led to the joint bestowal of the Turing Award on the two, for the introduction of this fundamental concept of computational complexity theory.

9.2 University of California, Berkeley, 1960–1963

Scott took up a post as Assistant Professor of Mathematics, back at the University of California, Berkeley, and involved himself with classical issues in mathematical logic, especially set theory and Tarskian model theory. During this period he started supervising Ph.D. students, such as James Halpern (Contributions to the Study of the Independence of the Axiom of Choice) and Edgar Lopez-Escobar (Infinitely Long Formulas with Countable Quantifier Degrees).

25 26 CHAPTER 9. DANA SCOTT

9.2.1 Modal and tense logic

Scott also began working on modal logic in this period, beginning a collaboration with John Lemmon, who moved to Claremont, California, in 1963. Scott was especially interested in Arthur Prior's approach to tense logic and the connection to the treatment of time in natural-language semantics, and began collaborating with Richard Montague (Copeland 2004), whom he had known from his days as an undergraduate at Berkeley. Later, Scott and Montague independently discovered an important generalisation of Kripke semantics for modal and tense logic, called Scott- Montague semantics (Scott 1970). John Lemmon and Scott began work on a modal-logic textbook that was interrupted by Lemmon’s death in 1966. Scott circulated the incomplete monograph amongst colleagues, introducing a number of important techniques in the semantics of model theory, most importantly presenting a refinement of canonical model that became standard, and introducing the technique of constructing models through filtrations, both of which are core concepts in modern Kripke semantics (Blackburn, de Rijke, and Venema, 2001). Scott eventually published the work as An Introduction to Modal Logic (Lemmon & Scott, 1977).

9.3 Stanford, Amsterdam and Princeton, 1963–1972

Following an initial observation of Robert Solovay, Scott formulated the concept of Boolean-valued model, as Solovay and Petr Vopěnka did likewise at around the same time. In 1967 Scott published a paper, A Proof of the Indepen- dence of the Continuum Hypothesis, in which he used Boolean-valued models to provide an alternate analysis of the independence of the continuum hypothesis to that provided by Paul Cohen. This work led to the award of the Leroy P. Steele Prize in 1972.

9.4 Oxford University, 1972–1981

Scott took up a post as Professor of Mathematical Logic on the Philosophy faculty of Oxford University in 1972. He was member of Merton College while at Oxford.

9.4.1 Semantics of programming languages

This period saw Scott working with Christopher Strachey, and the two managed, despite administrative pressures, to do work on providing a mathematical foundation for the semantics of programming languages, the work for which Scott is best known. Together, their work constitutes the Scott-Strachey approach to denotational semantics; it con- stitutes one of the pieces of work in theoretical computer science and can perhaps be regarded as founding one of the schools of computer science. One of Scott’s contributions is his formulation of domain theory, allowing programs involving recursive functions and looping-control constructs to be given denotational semantics. Additionally, he provided a foundation for the understanding of infinitary and continuous information through domain theory and his theory of information systems. Scott’s work of this period led to the bestowal of:

• The 1990 Harold Pender Award for his application of concepts from logic and algebra to the development of mathematical semantics of programming languages;

• The 1997 Rolf Schock Prize in logic and philosophy from the Royal Swedish Academy of Sciences for his conceptually oriented logical works, especially the creation of domain theory, which has made it possible to extend Tarski’s semantical paradigm to programming languages as well as to construct models of Curry’s combinatory logic and Church’s calculus of lambda conversion; and

• The 2001 Bolzano Prize for Merit in the Mathematical Sciences by the Czech Academy of Sciences.

• The 2007 EATCS Award for his contribution to theoretical computer science. 9.5. CARNEGIE MELLON UNIVERSITY 1981–2003 27

9.5 Carnegie Mellon University 1981–2003

At Carnegie Mellon University, Scott proposed the theory of equilogical spaces as a successor theory to domain theory; among its many advantages, the category of equilogical spaces is a cartesian closed category, whereas the category of domains[4] is not. In 1994, he was inducted as a Fellow of the Association for Computing Machinery. In 2012 he became a fellow of the American Mathematical Society.[5]

9.6 See also

• Scott’s trick

9.7 Bibliography

9.7.1 Works by Scott

• With Michael O. Rabin, 1959. Finite Automata and Their Decision Problem.

• 1967. A proof of the independence of the continuum hypothesis. Mathematical Systems Theory 1:89–111.

• 1970. 'Advice in modal logic'. In Philosophical Problems in Logic, ed. K. Lambert, pages 143–173.

• With John Lemmon, 1977. An Introduction to Modal Logic. Oxford: Blackwell.

9.7.2 Other works

• Blackburn, de Rijke and Venema (2001). Modal logic. Cambridge University Press.

• Jack Copeland (2004). Arthur Prior. In the Stanford Encyclopedia of Philosophy.

• Solomon Feferman and Anita Burdman Feferman (2004). Alfred Tarski: life and logic. Cambridge University Press, ISBN 0-521-80240-7, ISBN 978-0-521-80240-6.

• Solomon Feferman (2005). Tarski’s influence on computer science. Proc. LICS'05. IEEE Press.

• Joseph E. Stoy (1977). Denotational Semantics: The Scott-Strachey Approach to Programming Language The- ory. MIT Press. ISBN 0-262-19147-4

9.8 References

[1] “Dana Stewart Scott”. Mathematics Genealogy Project. North Dakota State University. Retrieved December 26, 2011.

[2] Feferman & Feferman 2004.

[3] Scott, Dana; Rabin, Michael (1959). “Finite Automata and Their Decision Problems”. IBM Journal of Research and Development 3 (2): 114–125. doi:10.1147/rd.32.0114.

[4] Where here Dana Scott counts the category of domains to be the category whose objects are pointed DCPOs, and whose morphisms are the strict, Scott-continuous functions

[5] List of Fellows of the American Mathematical Society, retrieved 2013-07-14. 28 CHAPTER 9. DANA SCOTT

9.9 External links

• Dana S. Scott home page

• DOMAIN 2002 Workshop on Domain Theory — held in honor of Scott’s 70th birthday. • Dana Scott at the Mathematics Genealogy Project

• List of publications from Microsoft Academic Search Chapter 10

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.

If δ is an ordinal number and ⟨Xα | α < δ⟩ is a sequence of subsets of δ , then the diagonal intersection, denoted by

∆α<δXα,

is defined to be

∩ {β < δ | β ∈ Xα}. α<β

That is, an ordinal β is in the diagonal intersection ∆α<δXα if and only if it is contained in the first β members of the sequence. This is the same as

∩ ([0, α] ∪ Xα), α<δ where the closed interval from 0 to α is used to avoid restricting the range of the intersection.

10.1 See also

• Fodor’s lemma • Club set

• Club filter

10.2 References

• , Set Theory, The Third Millennium Edition, Springer-Verlag Berlin Heidelberg New York, 2003, page 92.

• Akihiro Kanamori, The Higher Infinite, Second Edition, Springer-Verlag Berlin Heidelberg, 2009, page 2.

This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Com- mons Attribution/Share-Alike License.

29 Chapter 11

Elementary equivalence

In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elemen- tarily equivalent if they satisfy the same first-order σ-sentences. If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-order σ-formula φ(a1, …, an) with parameters a1, …, an from N is true in N if and only if it is true in M. If N is an elementary substructure of M, M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M. A substructure N of M is elementary if and only if it passes the Tarski–Vaught test: every first-order formula φ(x, b1, …, bn) with parameters in N that has a solution in M also has a solution in N when evaluated in M. One can prove that two structures are elementary equivalent with the Ehrenfeucht–Fraïssé games.

11.1 Elementarily equivalent structures

Two structures M and N of the same signature σ are elementarily equivalent if every first-order sentence (formula without free variables) over σ is true in M if and only if it is true in N, i.e. if M and N have the same complete first-order theory. If M and N are elementarily equivalent, one writes M ≡ N. A first-order theory is complete if and only if any two of its models are elementarily equivalent. For example, consider the language with one binary relation symbol '<'. The model R of real numbers with its usual order and the model Q of rational numbers with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense linear ordering. This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by Vaught’s test. More generally, any first-order theory has non-isomorphic, elementary equivalent models, which can be obtained via the Löwenheim–Skolem theorem. Thus, for example, there are non-standard models of Peano arithmetic, which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard model.

11.2 Elementary substructures and elementary extensions

N is an elementary substructure of M if N and M are structures of the same signature σ such that for all first-order σ-formulas φ(x1, …, xn) with free variables x1, …, xn, and all elements a1, …, a of N, φ(a1, …, a) holds in N if and only if it holds in M:

N |= φ(a1, …, an) iff M |= φ(a1, …, an).

It follows that N is a substructure of M. If N is a substructure of M, then both N and M can be interpreted as structures in the signature σN consisting of σ together with a new constant symbol for every element of N. N is an elementary substructure of M if and only if N is a substructure of M and N and M are elementarily equivalent as σN-structures.

30 11.3. TARSKI–VAUGHT TEST 31

If N is an elementary substructure of M, one writes N ⪯ M and says that M is an elementary extension of N: M ⪰ N. The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.

11.3 Tarski–Vaught test

The Tarski–Vaught test (or Tarski–Vaught criterion) is a necessary and sufficient condition for a substructure N of a structure M to be an elementary substructure. It can be useful for constructing an elementary substructure of a large structure. Let M be a structure of signature σ and N a substructure of M. N is an elementary substructure of M if and only if for every first-order formula φ(x, y1, …, yn) over σ and all elements b1, …, bn from N, if M |= x φ(x, b1, …, bn), then there is an element a in N such that M |= φ(a, b1, …, bn).

11.4 Elementary embeddings

An elementary embedding of a structure N into a structure M of the same signature σ is a h: N → M such that for every first-order σ-formula φ(x1, …, xn) and all elements a1, …, a of N,

N |= φ(a1, …, an) if and only if M |= φ(h(a1), …, h(an)).

Every elementary embedding is a strong homomorphism, and its image is an elementary substructure. Elementary embeddings are the most important maps in model theory. In set theory, elementary embeddings whose domain is V (the universe of set theory) play an important role in the theory of large cardinals (see also critical point).

11.5 References

• Chang, Chen Chung; Keisler, H. Jerome (1990) [1973], Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3.

• Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521- 58713-6. • Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, New York • Heidelberg • Berlin: Springer Verlag, ISBN 0-387-90170-1 Chapter 12

Equiconsistency

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, “as consistent as each other”. In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believed to be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistent—if we can do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T are equiconsistent.

12.1 Consistency

In mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own consistency. Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, the program quickly became the establishment of the consistency of arithmetic by methods formalizable within arithmetic itself. Gödel's incompleteness theorems show that Hilbert’s program cannot be realized: If a consistent recursively enumer- able theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (Robinson arithmetic suffices), then the theory cannot prove its own consistency. There are some technical caveats as to what requirements the formal statement representing the meta- mathematical statement “The theory is consistent” needs to satisfy, but the outcome is that if a (sufficiently strong) theory can prove its own consistency then either there is no computable way of identifying whether a statement is even an axiom of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, including false statements such as its own consistency). Given this, instead of outright consistency, one usually considers relative consistency: Let S and T be formal theories. Assume that S is a consistent theory. Does it follow that T is consistent? If so, then T is consistent relative to S. Two theories are equiconsistent if each one is consistent relative to the other.

12.2 Consistency strength

If T is consistent relative to S, but S is not known to be consistent relative to T, then we say that S has greater consistency strength than T. When discussing these issues of consistency strength the metatheory in which the discussion takes places needs to be carefully addressed. For theories at the level of second-order arithmetic, the reverse mathematics program has much to say. Consistency strength issues are a usual part of set theory, since this is a recursive theory that can certainly model most of mathematics. The usual set of axioms of set theory is called ZFC. When a set theoretic statement A is said to be equiconsistent to another B, what is being claimed is that in the metatheory (Peano Arithmetic in this case) it can be proven that the theories ZFC+A and ZFC+B are equiconsistent. Usually, primitive recursive arithmetic can be adopted as the metatheory in question, but even if the metatheory is

32 12.3. SEE ALSO 33

ZFC (for and with Zermelo’s axiom of choice) or an extension of it, the notion is meaningful. Thus, the method of forcing allows one to show that the theories ZFC, ZFC+CH and ZFC+¬CH are all equiconsistent. When discussing fragments of ZFC or their extensions (for example, ZF, set theory without the axiom of choice, or ZF+AD, set theory with the axiom of determinacy), the notions described above are adapted accordingly. Thus, ZF is equiconsistent with ZFC, as shown by Gödel. The consistency strength of numerous combinatorial statements can be calibrated by large cardinals. For example, the negation of Kurepa’s hypothesis is equiconsistent with an inaccessible cardinal, the non-existence of special ω2 - Aronszajn trees is equiconsistent with a , and the non-existence of ω2 -Aronszajn trees is equiconsistent with a .[1]

12.3 See also

• Large cardinal property

12.4 References

[1] • Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, p. 225, ISBN 978-1-84890- 050-9, Zbl 1262.03001

• Akihiro Kanamori (2003). The Higher Infinite. Springer. ISBN 3-540-00384-3 Chapter 13

Extender (set theory)

In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender. A (κ, λ)-extender can be defined as an elementary embedding of some model M of ZFC− (ZFC minus the power set axiom) having critical point κ ε M, and which maps κ to an ordinal at least equal to λ. It can also be defined as a collection of ultrafilters, one for each n-tuple drawn from λ.

13.1 Formal definition of an extender

<ω Let κ and λ be cardinals with κ≤λ. Then, a set E = {Ea|a ∈ [λ] } is called a (κ,λ)-extender if the following properties are satisfied:

1. each Ea is a κ-complete nonprincipal ultrafilter on [κ]<ω and furthermore

(a) at least one Ea is not κ+-complete, (b) for each α ∈ κ , at least one Ea contains the set {s ∈ [κ]|a| : α ∈ s} .

2. (Coherence) The Ea are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).

|a| |b| 3. (Normality) If f is such that {s ∈ [κ] : f(s) ∈ max s} ∈ Ea , then for some b ⊇ a, {t ∈ κ : (f ◦ πba)(t) ∈ t} ∈ Eb .

4. (Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of the ultrapowers Ult(V,Ea)).

By coherence, one means that if a and b are finite subsets of λ such that b is a superset of a, then if X is an element of the ultrafilter Eb and one chooses the right way to project X down to a set of sequences of length |a|, then X is { } ··· { } an element of Ea. More formally, for b = α1, . . . , αn , where α1 < < αn < λ , and a = αi1 , . . . , αim , where m≤n and for j≤m the ij are pairwise distinct and at most n, we define the projection πba : {ξ1, . . . , ξn} 7→ { } ··· ξi1 , . . . , ξim (ξ1 < < ξn) . Then Ea and Eb cohere if

X ∈ Ea ⇔ {s : πba(s) ∈ X} ∈ Eb

13.2 Defining an extender from an elementary embedding

Given an elementary embedding j:V→M, which maps the set-theoretic universe V into a transitive inner model M, <ω with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines E = {Ea|a ∈ [λ] } as follows:

34 13.3. REFERENCES 35

<ω <ω fora ∈ [λ] ,X ⊆ [κ] : X ∈ Ea ⇔ a ∈ j(X).

One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.

13.3 References

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.). Springer. ISBN 3-540-00384-3.

• Jech, Thomas (2002). Set Theory (3rd ed ed.). Springer. ISBN 3-540-44085-2. Chapter 14

Fodor’s lemma

In mathematics, particularly in set theory, Fodor’s lemma states the following: If κ is a regular, uncountable cardinal, S is a stationary subset of κ , and f : S → κ is regressive (that is, f(α) < α for any α ∈ S , α ≠ 0 ) then there is some γ and some stationary S0 ⊆ S such that f(α) = γ for any α ∈ S0 . In modern parlance, the nonstationary is normal.

14.1 Proof

We can assume that 0∈ / S (by removing 0, if necessary). If Fodor’s lemma is false, for every α < κ there is some −1 club set Cα such that Cα ∩ f (α) = ∅ . Let C = ∆α<κCα . The club sets are closed under diagonal intersection, so C is also club and therefore there is some α ∈ S ∩ C . Then α ∈ Cβ for each β < α , and so there can be no β < α such that α ∈ f −1(β) , so f(α) ≥ α , a contradiction. The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1956. It is sometimes also called “The Pressing Down Lemma”. Fodor’s lemma also holds for Thomas Jech's notion of stationary sets as well as for the general notion of stationary set.

14.2 Fodor’s lemma for trees

Another related statement, also known as Fodor’s lemma (or Pressing-Down-lemma), is the following: For every non-special tree T and regressive mapping f : T → T (that is, f(t) < t , with respect to the order on T , for every t ∈ T ), there is a non-special subtree S ⊂ T on which f is constant.

14.3 References

• G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. Szeged, 17(1956), 139- 142. • Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3. • Mark Howard, Applications of Fodor’s Lemma to Vaught’s Conjecture. Ann. Pure and Appl. Logic 42(1): 1-19 (1989). • Simon Thomas, The Automorphism Tower Problem. PostScript file at • S. Todorcevic, Combinatorial dichotomies in set theory. pdf at

This article incorporates material from Fodor’s lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

36 Chapter 15

Huge cardinal

In mathematics, a cardinal number κ is called huge if there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and

j(κ)M ⊂ M.

Here, αM is the class of all sequences of length α whose elements are in M. Huge cardinals were introduced by Kenneth Kunen (1978).

15.1 Variants

In what follows, jn refers to the n-th iterate of the elementary embedding j, that is, j composed with itself n times, for a finite ordinal n. Also, <αM is the class of all sequences of length less than α whose elements are in M. Notice that for the “super” versions, γ should be less than j(κ), not jn(κ) . κ is n-huge if and only if there is j : V → M with critical point κ and

n

κ is super almost n-huge if and only if for every ordinal γ there is j : V → M with critical point κ, γ

n

κ is n-huge if and only if there is j : V → M with critical point κ and

n j (κ)M ⊂ M.

κ is super n-huge if and only if for every ordinal γ there is j : V → M with critical point κ, γ

n j (κ)M ⊂ M.

Notice that 0-huge is the same as measurable cardinal; and 1-huge is the same as huge. A cardinal satisfying one of the rank into rank axioms is n-huge for all finite n. The existence of an almost huge cardinal implies that Vopenka’s principle is consistent; more precisely any almost huge cardinal is also a Vopenka cardinal.

37 38 CHAPTER 15. HUGE CARDINAL

15.2 Consistency strength

The cardinals are arranged in order of increasing consistency strength as follows:

• almost n-huge • super almost n-huge

• n-huge

• super n-huge • almost n+1-huge

The consistency of a huge cardinal implies the consistency of a supercompact cardinal, nevertheless, the least huge cardinal is smaller than the least supercompact cardinal (assuming both exist).

15.3 ω-huge cardinals

One can try defining an ω-huge cardinal κ as one such that an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and λM⊆M, where λ is the supremum of jn(κ) for positive integers n. However Kunen’s inconsistency theorem shows that ω-huge cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF.

15.4 See also

• List of large cardinal properties

• The Dehornoy order on a braid group was motivated by properties of huge cardinals.

15.5 References

• Kunen, Kenneth (1978), “Saturated ideals”, The Journal of Symbolic Logic 43 (1): 65–76, doi:10.2307/2271949, ISSN 0022-4812, MR 495118 • Penelope Maddy,"Believing the Axioms,II"(i.e. part 2 of 2),"Journal of Symbolic Logic”,vol.53,no.3,Sept.1988,pages 736 to 764 (esp.754-756). • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.). Springer. ISBN 3-540-00384-3. Chapter 16

Inaccessible cardinal

In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak , and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case ℵ0 is strongly inaccessible). Weakly inaccessible cardinals were introduced by Hausdorff (1908), and strongly inaccessible ones by Sierpiński & Tarski (1930) and Zermelo (1930). The term “inaccessible cardinal” is ambiguous. Until about 1950 it meant “weakly inaccessible cardinal”, but since then it usually means “strongly inaccessible cardinal”. Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuum hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.

ℵ0 (aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal num- ber is either regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible. An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and ℵ0 are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.

16.1 Models and consistency

ZFC implies that the Vκ is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that the Gödel universe Lκ is a model of ZFC whenever κ is weakly inaccessible. Thus ZF together with “there exists a weakly inaccessible cardinal” implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal. If V is a standard model of ZFC and κ is an inaccessible in V, then: Vκ is one of the intended models of Zermelo– Fraenkel set theory; and Def (Vκ) is one of the intended models of Von Neumann–Bernays–Gödel set theory; and Vκ₊₁ is one of the intended models of Morse–Kelley set theory. Here Def (X) is the Δ0 definable subsets of X (see ). However, κ does not need to be inaccessible, or even a cardinal number, in order for Vκ to be a standard model of ZF (see below). Suppose V is a model of ZFC. Either V contains no strong inaccessible or, taking κ to be the smallest strong inac- cessible in V, Vκ is a standard model of ZFC which contains no strong inaccessibles. Thus, the consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles”. Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then Lκ is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles”. This shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals. The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched

39 40 CHAPTER 16. INACCESSIBLE CARDINAL

in the previous paragraph that the consistency of ZFC + “there is an inaccessible cardinal” implies the consistency of ZFC + “there is not an inaccessible cardinal” can be formalized in ZFC. However, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + “there is an inaccessible cardinal” can be formalized in ZFC. This follows from Gödel’s second incompleteness theorem, which shows that if ZFC + “there is an inaccessible cardinal” is consistent, then it cannot prove its own consistency. Because ZFC + “there is an inaccessible cardinal” does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + “there is an inaccessible cardinal” then this latter theory would be able to prove its own consistency, which is impossible if it is consistent. There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by Hrbacek & Jech (1999, p. 279), is that the class of all ordinals of a particular model M of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending M.

16.2 Existence of a proper class of inaccessibles

There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger, μ < κ. Thus this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (which could be confused with ZFC with urelements). This is useful to prove for example that every category has an appropriate Yoneda embedding. This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e. the class of all ordinals in your model.

16.3 α-inaccessible cardinals and hyper-inaccessible cardinals

A cardinal κ is α-inaccessible, for α any ordinal, if and only if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular). The α-inaccessible cardinals can be equivalently described as fixed points of functions which count the lower inacces- th sibles. For example, denote by ψ0(λ) the λ inaccessible cardinal, then the fixed points of ψ0 are the 1-inaccessible cardinals. Then letting ψᵦ(λ) be the λth β-inaccessible cardinal, the fixed points of ψᵦ are the (β+1)-inaccessible cardinals (the values ψᵦ₊₁(λ)). If α is a limit ordinal, an α-inaccessible is a fixed point of every ψᵦ for β < α (the value ψα(λ) is the λth such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers. The term hyper-inaccessible is ambiguous. Some authors use it to mean 1-inaccessible, though this use is rare. Most authors use it to mean that κ is κ-inaccessible. (It can never be κ+1-inaccessible.) For any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ. Hyper-hyper-inaccessible cardinals and so on can be defined in a similar way. Using “weakly inaccessible” instead of “inaccessible”, similar definitions can be made for “weakly α-inaccessible”, “weakly hyper-inaccessible”, and “weakly α-hyper-inaccessible”. Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.

16.4 Two model-theoretic characterisations of inaccessibility

Firstly, a cardinal κ is inaccessible if and only if κ has the following reflection property: for all subsets U ⊂ Vκ, there exists α < κ such that (Vα, ∈,U ∩ Vα) is an elementary substructure of (Vκ, ∈,U) . (In fact, the set of such α is 0 closed unbounded in κ.) Equivalently, κ is Πn -indescribable for all n ≥ 0. 16.5. SEE ALSO 41

It is provable in ZF that ∞ satisfies a somewhat weaker reflection property, where the substructure (Vα, ∈, U ∩ Vα) is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation |= can be defined, truth itself cannot, due to Tarski’s theorem. Secondly, under ZFC it can be shown that κ is inaccessible if and only if (Vκ, ∈) is a model of second order ZFC. In this case, by the reflection property above, there exists α < κ such that (Vα, ∈) is a standard model of (first order) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a standard model of ZFC.

16.5 See also

• Mahlo cardinal

• Club set • Inner model

• • Constructible universe

16.6 References

• Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, Elsevier Science Ltd, ISBN 0-444-10535-2 • Hausdorff, Felix (1908), “Grundzüge einer Theorie der geordneten Mengen”, Mathematische Annalen 65 (4): 435–505, doi:10.1007/BF01451165, ISSN 0025-5831 • Hrbacek, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 978-0- 8247-7915-3 • Kanamori, Akihiro (2003), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.), Springer, ISBN 3-540-00384-3 • Sierpiński, Wacław; Tarski, Alfred (1930), “Sur une propriété caractéristique des nombres inaccessibles”, Fundamenta Mathematicae 15: 292–300, ISSN 0016-2736 • Zermelo, Ernst (1930), "Über Grenzzablen und Mengenbereiche”, Fundamenta Mathematicae 16: 29–47, ISSN 0016-2736 Chapter 17

Ineffable cardinal

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). A cardinal number κ is called almost ineffable if for every f : κ → P(κ) (where P(κ) is the powerset of κ ) with the property that f(δ) is a subset of δ for all ordinals δ < κ , there is a subset S of κ having cardinal κ and homogeneous for f , in the sense that for any δ1 < δ2 in S , f(δ1) = f(δ2) ∩ δ1 .

A cardinal number κ is called ineffable if for every binary-valued function f : P=2(κ) → {0, 1} , there is a stationary subset of κ on which f is homogeneous: that is, either f maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one.

More generally, κ is called n -ineffable (for a positive n ) if for every f : P=n(κ) → {0, 1} there is a stationary subset of κ on which f is n -homogeneous (takes the same value for all unordered n - drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable.

A totally ineffable cardinal is a cardinal that is n -ineffable for every 2 ≤ n < ℵ0 . If κ is (n + 1) -ineffable, then the set of n -ineffable cardinals below κ is a stationary subset of κ . Totally ineffable cardinals are of greater consistency strength than subtle cardinals and of lesser consistency strength than remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.

17.1 References

• Friedman, Harvey (2001), “Subtle cardinals and linear orderings”, Annals of Pure and Applied Logic 107 (1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1. • Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript

42 Chapter 18

Intersection (set theory)

A∩B BA

Intersection of two sets: A ∩ B

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.[1] For explanation of the symbols used in this article, refer to the table of mathematical symbols.

18.1 Basic definition

The intersection of A and B is written "A ∩ B". Formally:

43 44 CHAPTER 18. INTERSECTION (SET THEORY)

Intersection of three sets: A ∩ B ∩ C

A ∩ B = {x : x ∈ A ∧ x ∈ B} that is

x ∈ A ∩ B if and only if

• x ∈ A and • x ∈ B.

For example:

• The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. • The number 9 is not in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of odd numbers {1, 3, 5, 7, 9, 11, …}.[2] 18.1. BASIC DEFINITION 45

Intersections of the Greek, English and Russian alphabet (upper case graphemes)

More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C. Inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. Now the intersection of A and B may be written as the complement of the union of their complements, derived easily from De Morgan’s laws: A ∩ B = (Ac ∪ Bc)c

18.1.1 Intersecting and disjoint sets

We say that A intersects (meets) B at an element x if x belongs to A and B. We say that A intersects (meets) B if A intersects B at some element. A intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B. In plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅ . For example, the sets {1, 2} and {3, 4} are disjoint, the set of even numbers intersects the set of multiples of 3 at 0, 6, 12, 18 and other numbers. 46 CHAPTER 18. INTERSECTION (SET THEORY)

Example of an intersection with sets

18.2 Arbitrary intersections

The most general notion is the intersection of an arbitrary nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, x is an element of A. In symbols:

( ∩ ) x ∈ M ⇔ (∀A ∈ M, x ∈ A) .

The notation for this last concept can vary considerably. Set theorists will sometimes write "⋂M", while others will instead write "⋂A∈MA". The latter notation can be generalized to "⋂i∈I Ai", which refers to the intersection of the collection {Ai : i ∈ I}. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of natural numbers, notation analogous to that of an infinite series may be 18.3. NULLARY INTERSECTION 47

seen:

∩∞ Ai. i=1

When formatting is difficult, this can also be written "A1 ∩ A2 ∩ A3 ∩ ...”, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩ ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for an example see the article on σ-algebras.) Finally, let us note that whenever the symbol "∩" is placed before other symbols instead of between them, it should be of a larger size (⋂).

18.3 Nullary intersection

Conjunctions of the arguments in parentheses The conjunction of no argument is the tautology (compare: empty product); accordingly the intersection of no set is the universe.

Note that in the previous section we excluded the case where M was the (∅). The reason is as follows: The intersection of the collection M is defined as the set (see set-builder notation)

∩ M = {x : ∀A ∈ M, x ∈ A}.

If M is empty there are no sets A in M, so the question becomes “which x's satisfy the stated condition?" The answer seems to be every possible x. When M is empty the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection) [3] 48 CHAPTER 18. INTERSECTION (SET THEORY)

Unfortunately, according to standard (ZFC) set theory, the universal set does not exist. A partial fix for this problem can be found if we agree to restrict our attention to subsets of a fixed set U called the universe. In this case the intersection of a family of subsets of U can be defined as

∩ M = {x ∈ U : ∀A ∈ M, x ∈ A}.

Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-defined set by assumption and becomes the identity element for this operation.

18.4 See also

• Complement

• Intersection graph • Logical conjunction

• Naive set theory • Symmetric difference

• Union • Cardinality

• Iterated • MinHash

18.5 References

[1] “Stats: Rules”. People.richland.edu. Retrieved 2012-05-08.

[2] How to find the intersection of sets

[3] Megginson, Robert E. (1998), “Chapter 1”, An introduction to theory, Graduate Texts in Mathematics 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3

18.6 Further reading

• Devlin, K.J., The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd edition, Springer-Verlag, New York, NY, 1993.

• “Chapter 1” Munkres, James R. Topology. 2nd edition. Upper Saddle River: Prentice Hall, 2000. • “Chapter 2”. Discrete Mathematics and Its Applications by Kenneth-Rosen. ISBN 978-0-07-322972-0.

18.7 External links

• Weisstein, Eric W., “Intersection”, MathWorld. Chapter 19

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional . For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume.[1] It is used throughout real analysis, in particular to define . Sets that can be assigned a Lebesgue measure are called Lebesgue measurable; the measure of the Lebesgue measurable set A is denoted by λ(A). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.[2] The Lebesgue measure is often denoted dx, but this should not be confused with the distinct notion of a volume form.

19.1 Definition

Given a subset E ⊆ R , with the length of an (open, closed, semi-open) interval I = [a, b] given by l(I) = b − a , the Lebesgue outer measure λ∗(E) is defined as

{ } ∑∞ ∪∞ ∗ λ (E) = inf l(Ik):(Ik)k∈N with intervals open disjoint of sequence a is E ⊆ Ik k=1 k=1

The Lebesgue measure of E is given by its Lebesgue outer measure λ(E) = λ∗(E) if, for every A ⊆ R ,

λ∗(A) = λ∗(A ∩ E) + λ∗(A ∩ Ec)

19.1.1 Intuition

The first part of the definition states that the subset E of the real numbers is reduced to its outer measure by coverage by sets of intervals. Each of these sets of intervals I covers E in the sense that when the intervals are combined together by union, they form a superset of E . Moreover, the intervals in each set are disjoint, and there is a countable infinity of these intervals. For each set, the total length is calculated by adding the lengths of this infinity of disjoint intervals. This total length of any interval set can easily overestimate the measure of E , because E is a subset of the union of the intervals, and so the intervals may include points which are not in E . The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E most tightly. That characterizes the Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition is tested by taking subsets of the real numbers A using E as an instrument to split A into two partitions: the part of A which intersects with E and the remaining part of A which is not in E : the set difference of A and E . These partitions of A are subject to the outer measure. If for

49 50 CHAPTER 19. LEBESGUE MEASURE all possible such subsets A of the real numbers, the partitions of A cut apart by E have outer measures which add up to the outer measure of A , then the outer Lebesgue measure of E gives its Lebesgue measure. Intuitively, this condition means that the set E must not have some curious properties which causes a discrepancy in the measure of another set when E is used as a “mask” to “clip” that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure. (Such sets are, in fact, not Lebesgue-measurable.)

19.2 Examples

• Any closed interval [a, b] of real numbers is Lebesgue measurable, and its Lebesgue measure is the length b−a. The open interval (a, b) has the same measure, since the difference between the two sets consists only of the end points a and b and has measure zero.

• Any Cartesian product of intervals [a, b] and [c, d] is Lebesgue measurable, and its Lebesgue measure is (b−a)(d−c), the area of the corresponding rectangle.

• Moreover, every Borel set is Lebesgue measurable. However, there are Lebesgue measurable sets which are not Borel sets.

• Any of real numbers has Lebesgue measure 0.

• In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R.

• The Cantor set is an example of an uncountable set that has Lebesgue measure zero.

• Vitali sets are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on the axiom of choice.

19.3 Properties

A={x+t | x∊A}

t

A

Translation invariance: The Lebesgue measure of A and A + t are the same.

The Lebesgue measure on Rn has the following properties: 19.4. NULL SETS 51

1. If A is a cartesian product of intervals I1 × I2 × ... × In, then A is Lebesgue measurable and λ(A) = |I1| · |I2| · · · |In|. Here, |I| denotes the length of the interval I. 2. If A is a of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue mea- surable and λ(A) is equal to the sum (or infinite series) of the measures of the involved measurable sets.

3. If A is Lebesgue measurable, then so is its complement.

4. λ(A) ≥ 0 for every Lebesgue measurable set A.

5. If A and B are Lebesgue measurable and A is a subset of B, then λ(A) ≤ λ(B). (A consequence of 2, 3 and 4.)

6. Countable unions and intersections of Lebesgue measurable sets are Lebesgue measurable. (Not a consequence of 2 and 3, because a that is closed under complements and disjoint countable unions need not be closed under countable unions: {∅, {1, 2, 3, 4}, {1, 2}, {3, 4}, {1, 3}, {2, 4}} .)

7. If A is an open or closed subset of Rn (or even Borel set, see ), then A is Lebesgue measurable.

8. If A is a Lebesgue measurable set, then it is “approximately open” and “approximately closed” in the sense of Lebesgue measure (see the regularity theorem for Lebesgue measure).

9. A Lebesgue-measurable set can be “squeezed” between a containing open set and a contained . I.e, if A is Lebesgue measurable then for every positive ε there exist an open set G and a closed set F such that G⊇A⊇F and λ(G\A)<ε and λ(A\F)<ε.

10. A Lebesgue-measurable set can be “squeezed” between a containing Gδ set and a contained Fσ set. I.e, if A is Lebesgue measurable then there exist a Gδ set G and an Fσ set F such that G⊇A⊇F and λ(G\A)=λ(A\F)=0.

11. Lebesgue measure is both locally finite and inner regular, and so it is a .

12. Lebesgue measure is strictly positive on non-empty open sets, and so its is the whole of Rn.

13. If A is a Lebesgue measurable set with λ(A) = 0 (a ), then every subset of A is also a null set. A fortiori, every subset of A is measurable.

14. If A is Lebesgue measurable and x is an element of Rn, then the translation of A by x, defined by A + x = {a + x : a ∈ A}, is also Lebesgue measurable and has the same measure as A.

15. If A is Lebesgue measurable and δ > 0 , then the dilation of A by δ defined by δA = {δx : x ∈ A} is also Lebesgue measurable and has measure δnλ (A).

16. More generally, if T is a linear transformation and A is a measurable subset of Rn, then T(A) is also Lebesgue measurable and has the measure | det(T )| λ (A) .

All the above may be succinctly summarized as follows:

The Lebesgue measurable sets form a σ-algebra containing all products of intervals, and λ is the unique complete translation-invariant measure on that σ-algebra with λ([0, 1] × [0, 1] × · · · × [0, 1]) = 1.

The Lebesgue measure also has the property of being σ-finite.

19.4 Null sets

Main article: Null set

A subset of Rn is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets. If a subset of Rn has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric on Rn (or any metric Lipschitz equivalent to it). On the other hand a set may have topological dimension less than n and have positive n-dimensional Lebesgue 52 CHAPTER 19. LEBESGUE MEASURE

measure. An example of this is the Smith–Volterra–Cantor set which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure. In order to show that a given set A is Lebesgue measurable, one usually tries to find a “nicer” set B which differs from A only by a null set (in the sense that the symmetric difference (A − B) ∪ (B − A) is a null set) and then show that B can be generated using countable unions and intersections from open or closed sets.

19.5 Construction of the Lebesgue measure

The modern construction of the Lebesgue measure is an application of Carathéodory’s extension theorem. It proceeds as follows. Fix n ∈ N.A box in Rn is a set of the form

∏n B = [ai, bi] , i=1 where bi ≥ ai, and the product symbol here represents a Cartesian product. The volume of this box is defined to be

∏n vol(B) = (bi − ai) . i=1 For any subset A of Rn, we can define its outer measure λ*(A) by:

{ ∑ } λ∗(A) = inf vol(B): C covers union whose boxes of collection countable a is A . B∈C We then define the set A to be Lebesgue measurable if for every subset S of Rn,

λ∗(S) = λ∗(S ∩ A) + λ∗(S \ A) . These Lebesgue measurable sets form a σ-algebra, and the Lebesgue measure is defined by λ(A) = λ*(A) for any Lebesgue measurable set A. The existence of sets that are not Lebesgue measurable is a consequence of a certain set-theoretical axiom, the axiom of choice, which is independent from many of the conventional systems of axioms for set theory. The Vitali theorem, which follows from the axiom, states that there exist subsets of R that are not Lebesgue measurable. Assuming the axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach–Tarski paradox. In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue measurable is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom of choice (see Solovay’s model).[3]

19.6 Relation to other measures

The agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not complete. The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure (Rn with addition is a locally compact group). The Hausdorff measure is a generalization of the Lebesgue measure that is useful for measuring the subsets of Rn of lower dimensions than n, like submanifolds, for example, surfaces or curves in R³ and fractal sets. The Hausdorff measure is not to be confused with the notion of Hausdorff dimension. It can be shown that there is no infinite-dimensional analogue of Lebesgue measure. 19.7. SEE ALSO 53

19.7 See also

• Lebesgue’s density theorem

• Lebesgue measure of the set of Liouville numbers

19.8 References

[1] The term volume is also used, more strictly, as a synonym of 3-dimensional volume

[2] Henri Lebesgue (1902). “Intégrale, longueur, aire”. Université de Paris.

[3] Solovay, Robert M. (1970). “A model of set-theory in which every set of reals is Lebesgue measurable”. Annals of Mathematics. Second Series 92 (1): 1–56. doi:10.2307/1970696. JSTOR 1970696. Chapter 20

Measurable cardinal

In mathematics, a measurable cardinal is a certain kind of large cardinal number.

20.1 Measurable

Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ of cardinality λ<κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals the sum of the measures of the individual Aα.) Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott’s trick. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter. Again, this means that the intersection of any strictly less than κ-many sets in the ultrafilter, is also in the ultrafilter. Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacy that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset. The concept of a measurable cardinal was introduced by Stanislaw Ulam (1930), who showed that the smallest car- dinal κ that admits a non-trivial countably-additive two-valued measure must in fact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible. It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2λ. If this were the case, then we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1. The intersection of these λ-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure. Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility. If κ is measurable and p∈Vκ and M (the ultrapower of V) satisfies ψ(κ,p), then the set of α<κ such that V satisfies ψ(α,p) is stationary in κ (actually a set of measure 1). In particular if ψ is a Π1 formula and V satisfies ψ(κ,p), then M satisfies it and thus V satisfies ψ(α,p) for a stationary set of α<κ. This property can be used to show that κ is a limit of most types of large cardinals which are weaker than measurable. Notice that the ultrafilter or measure which witnesses that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it which is impossible.

54 20.2. REAL-VALUED MEASURABLE 55

Every measurable cardinal κ is a 0-huge cardinal because κM⊂M, that is, every function from κ to M is in M. Con- sequently, Vκ₊₁⊂M.

20.2 Real-valued measurable

A cardinal κ is called real-valued measurable if there is a κ-additive on the power set of κ which vanishes on singletons. Real-valued measurable cardinals were introduced by Stefan Banach (1930). Banach & Kuratowski (1929) showed that the continuum hypothesis implies that c is not real-valued measurable. Stanislaw Ulam (1930) showed that real valued measurable cardinals are weakly inaccessible (they are in fact weakly Mahlo). All measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ is measurable if and only if κ is greater than c . Thus a cardinal is measurable if and only if it is real-valued measurable and strongly inaccessible. A real valued measurable cardinal less than or equal to c exists if and only if there is a countably additive extension of the Lebesgue measure to all sets of real numbers if and only if there is an atomless probability measure on the power set of some non-empty set. Solovay (1971) showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are equiconsistent.

20.3 See also

• Normal measure

• Mitchell order

20.4 References

• Banach, Stefan (1930), "Über additive Maßfunktionen in abstrakten Mengen”, Fundamenta Mathematicae 15: 97–101, ISSN 0016-2736 • Banach, Stefan; Kuratowski, C. (1929), “Sur une généralisation du probleme de la mesure”, Fundamenta Math- ematicae 14: 127–131, ISSN 0016-2736 • Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 978-0-7204-2279-5.

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.). Springer. ISBN 3-540-00384-3.

• Solovay, Robert M. (1971), “Real-valued measurable cardinals”, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), Providence, R.I.: Amer. Math. Soc., pp. 397–428, MR 0290961 • Ulam, Stanislaw (1930), “Zur Masstheorie in der allgemeinen Mengenlehre”, Fundamenta Mathematicae 16: 140–150, ISSN 0016-2736 Chapter 21

Measure (mathematics)

For the coalgebra concept, see measuring coalgebra. In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word, specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the “smaller” subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.[1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the . Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

21.1 Definition

Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:

• Non-negativity: For all E in Σ: μ(E) ≥ 0.

• Null empty set: μ(∅) = 0.

• { } Countable additivity (or σ-additivity): For all countable collections Ei i∈N of pairwise disjoint sets in Σ: ( ) ∪ ∑ µ Ei = µ (Ei) i∈N i∈N

56 21.2. EXAMPLES 57

One may require that at least one set E has finite measure. Then the empty set automatically has measure zero because of countable additivity, because µ(E) = µ(E ∪ ∅) = µ(E) + µ(∅) , so µ(∅) = µ(E) − µ(E) = 0 . If only the second and third conditions of the definition of measure above are met, and μ takes on at most one of the values ±∞, then μ is called a .

The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets. If (X, ΣX ) and (Y, ΣY ) are two measurable spaces, then a function f : X → Y is called measurable if for every Y-measurable set B ∈ ΣY , (−1) the inverse image is X-measurable – i.e.: f (B) ∈ ΣX . The composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. A triple (X, Σ, μ) is called a measure space.A probability measure is a measure with total measure one – i.e. μ(X) = 1. A is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.

21.2 Examples

Some important measures are listed here.

• The counting measure is defined by μ(S) = number of elements in S.

• The Lebesgue measure on R is a complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0, 1]) = 1; and every other measure with these properties extends Lebesgue measure.

• Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.

• The Haar measure for a locally compact is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.

• The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.

• Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure. See .

• The Dirac measure δa (cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and strong measure zero. In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see “generalizations” below. Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics. Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.

21.3 Properties

Several further properties can be derived from the definition of a countably additive measure. 58 CHAPTER 21. MEASURE (MATHEMATICS)

21.3.1 Monotonicity

A measure μ is monotonic: If E1 and E2 are measurable sets with E1 ⊆ E2 then

µ(E1) ≤ µ(E2).

21.3.2 Measures of infinite unions of measurable sets

A measure μ is countably subadditive: For any countable sequence E1, E2, E3, ... of sets En in Σ (not necessarily disjoint):

( ) ∪∞ ∑∞ µ Ei ≤ µ(Ei). i=1 i=1

A measure μ is continuous from below: If E1, E2, E3, ... are measurable sets and En is a subset of En ₊ ₁ for all n, then the union of the sets En is measurable, and

( ) ∪∞ µ Ei = lim µ(Ei). i→∞ i=1

21.3.3 Measures of infinite intersections of measurable sets

A measure μ is continuous from above: If E1, E2, E3, ..., are measurable sets and for all n, En ₊ ₁ ⊂ En, then the intersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then

( ) ∩∞ µ Ei = lim µ(Ei). i→∞ i=1

This property is false without the assumption that at least one of the En has finite measure. For instance, for each n ∈ N, let En = [n, ∞) ⊂ R, which all have infinite Lebesgue measure, but the intersection is empty.

21.4 Sigma-finite measures

Main article: Sigma-finite measure

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure 1 µ(X) µ . A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure. For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k, k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'. 21.5. COMPLETENESS 59

21.5 Completeness

Main article:

A measurable set X is called a null set if μ(X) = 0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable. A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

21.6 Additivity

Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I and any set of nonnegative ri, i ∈ I define:

{ } ∑ ∑ ri = sup ri : |J| < ℵ0,J ⊆ I . i∈I i∈J That is, we define the sum of the ri to be the supremum of all the sums of finitely many of them.

A measure μ on Σ is κ-additive if for any λ < κ and any family Xα , α < λ the following hold:

∪ Xα ∈ Σ α∈λ ( ) ∪ ∑ µ Xα = µ (Xα) . α∈λ α∈λ Note that the second condition is equivalent to the statement that the ideal of null sets is κ-complete.

21.7 Non-measurable sets

Main article: Non-measurable set

If the axiom of choice is assumed to be true, not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach– Tarski paradox.

21.8 Generalizations

For certain purposes, it is useful to have a “measure” whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a . Measures that take values in Banach spaces have been studied extensively.[2] A measure that takes values in the set of self-adjoint pro- jections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under but not general , while signed measures are the linear closure of positive measures. Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this 60 CHAPTER 21. MEASURE (MATHEMATICS) definition was used first. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L∞ and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. A charge is a generalization in both directions: it is a finitely additive, signed measure.

21.9 See also

• Abelian von Neumann algebra

• Almost everywhere

• Carathéodory’s extension theorem

• Fubini’s theorem

• Fatou’s lemma

• Fuzzy measure theory

• Geometric measure theory

• Hausdorff measure

• Inner measure

• Lebesgue integration

• Lebesgue measure

• Lorentz space

• Lifting theory

• Measurable function

• Outer measure

• Product measure

• Pushforward measure

•

• Volume form

• Measurable cardinal

21.10 References

[1] Halmos, Paul (1950), Measure theory, Van Nostrand and Co.

[2] Rao, M. M. (2012), Random and vector measures, Series on Multivariate Analysis 9, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, ISBN 978-981-4350-81-5, MR 2840012. 21.11. BIBLIOGRAPHY 61

21.11 Bibliography

• Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience.

• Bauer, H. (2001), Measure and Integration Theory, Berlin: de Gruyter, ISBN 978-3110167191 • Bear, H.S. (2001), A Primer of Lebesgue Integration, San Diego: Academic Press, ISBN 978-0120839711

• Bogachev, V. I. (2006), Measure theory, Berlin: Springer, ISBN 978-3540345138

• Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III. • R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.

• Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160 Second edition.

• D. H. Fremlin, 2000. Measure Theory. Torres Fremlin. • Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer Verlag, ISBN 3-540-44085-2 • R. Duncan Luce and Louis Narens (1987). “measurement, theory of,” The New Palgrave: A Dictionary of Economics, v. 3, pp. 428–32. • M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.

• K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983), Theory of Charges: A Study of Finitely Additive Measures, London: Academic Press, pp. x + 315, ISBN 0-12-095780-9

• Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.

• Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes) • Terence Tao, 2011. An Introduction to Measure Theory. American Mathematical Society.

• Nik Weaver, 2013. Measure Theory and Functional Analysis. World Scientific Publishing.

21.12 External links

• Hazewinkel, Michiel, ed. (2001), “Measure”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4

• Tutorial: Measure Theory for Dummies 62 CHAPTER 21. MEASURE (MATHEMATICS)

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. 21.12. EXTERNAL LINKS 63

μ( ) =μ( ) + μ( ) +μ( ) + ...

Countable additivity of a measure μ: The measure of a countable disjoint union is the same as the sum of all measures of each subset. Chapter 22

Mitchell order

In mathematical set theory, the Mitchell order is a well-founded preorder on the set of normal measures on a measurable cardinal κ. It is named for William Mitchell. We say that M ◅ N (this is a strict order) if M is in the ultrapower model defined by N. Intuitively, this means that M is a weaker measure than N (note, for example, that κ will still be measurable in the ultrapower for N, since M is a measure on it). In fact, the Mitchell order can be defined on the set (or proper class, as the case may be) of extenders for κ; but if it is so defined it may fail to be transitive, or even well-founded, provided κ has sufficiently strong large cardinal properties. Well-foundedness fails specifically for rank-into-rank extenders; but Itay Neeman showed in 2004 that it holds for all weaker types of extender. The Mitchell rank of a measure is the ordertype of its predecessors under ◅; since ◅ is well-founded this is always an ordinal. A cardinal which has measures of Mitchell rank α for each α < β is said to be β-measurable.

22.1 References

• John Steel (Sep 1993). “The Well-Foundedness of the Mitchell Order”. Journal of Symbolic Logic 58 (3): 931–940. doi:10.2307/2275105.

• Itay Neeman (2004). “The Mitchell order below rank-to-rank”. J. Symbolic Logic 69 (4): 1143–1162. doi:10.2178/jsl/1102022215.

• Akihiro Kanamori (1997). The higher infinite. Perspectives in Mathematical Logic. Springer. • Donald A. Martin and John Steel (1994). “Iteration trees”. Journal of the Americal Mathematical Society 7: 1–73. • William Mitchell (1974). “Sets constructible from sequences of ultrafilters”. Journal of Symbolic Logic 39: 57–66. doi:10.2307/2272343.

64 Chapter 23

Model theory

This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model.

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a . We call a theory a set of sentences in a formal language, and model of a theory a structure (e.g. an ) that satisfies the sentences of that theory. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1]

universal algebra + logic = model theory.

Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997):

model theory = algebraic geometry − fields, although model theorists are also interested in the study of fields. Other nearby areas of mathematics include combinatorics, number theory, arithmetic dynamics, analytic functions, and non-standard analysis. In a similar way to , model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. The most prominent professional organization in the field of model theory is the Association for Symbolic Logic.

23.1 Branches of model theory

This article focuses on finitary first order model theory of infinite structures. Finite model theory, which concentrates on finite structures, diverges significantly from the study of infinite structures in both the problems studied and the techniques used. Model theory in higher-order or infinitary logics is hampered by the fact that completeness does not in general hold for these logics. However, a great deal of study has also been done in such languages. Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic. Examples of early theorems from classical model theory include Gödel’s completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of non-standard analysis. An important step in the evolution of classical model theory occurred with

65 66 CHAPTER 23. MODEL THEORY the birth of stability theory (through Morley’s theorem on uncountably categorical theories and Shelah's classification program), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. The result of this synthesis is called geometric model theory in this article (which is taken to include o-minimality, for example, as well as classical geometric stability theory). An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields. The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory.

23.2 Universal algebra

Main article: Universal algebra

Fundamental concepts in universal algebra are signatures σ and σ-algebras. Since these concepts are formally defined in the article on structures, the present article can content itself with an informal introduction which consists in examples of how these terms are used.

The standard signature of rings is σᵣᵢ = {×,+,−,0,1}, where × and + are binary, − is unary, and 0 and 1 are nullary. The standard signature of semirings is σᵣ = {×,+,0,1}, where the arities are as above. The standard signature of groups (with multiplicative notation) is σᵣ = {×,−1,1}, where × is binary, −1 is unary and 1 is nullary. The standard signature of monoids is σ = {×,1}. A ring is a σᵣᵢ-structure which satisfies the identities u + (v + w) = (u + v) + w, u + v = v + u, u + 0 = u, u + (−u) = 0, u × (v × w) = (u × v) × w, u × 1 = u, 1 × u = u, u × (v + w) = (u × v) + (u × w) and (v + w) × u = (v × u) + (w × u). A group is a σᵣ-structure which satisfies the identities u × (v × w) = (u × v) × w, u × 1 = u, 1 × u = u, u × u−1 = 1 and u−1 × u = 1. A monoid is a σ-structure which satisfies the identities u × (v × w) = (u × v) × w, u × 1 = u and 1 × u = u. A semigroup is a {×}-structure which satisfies the identity u × (v × w) = (u × v) × w. A magma is just a {×}-structure.

This is a very efficient way to define most classes of algebraic structures, because there is also the concept of σ- homomorphism, which correctly specializes to the usual notions of homomorphism for groups, semigroups, magmas and rings. For this to work, the signature must be chosen well. Terms such as the σᵣᵢ-term t(u,v,w) given by (u + (v × w)) + (−1) are used to define identities t = t ' , but also to construct free algebras. An equational class is a class of structures which, like the examples above and many others, is defined as the class of all σ-structures which satisfy a certain set of identities. Birkhoff’s theorem states:

A class of σ-structures is an equational class if and only if it is not empty and closed under subalgebras, homomorphic images, and direct products.

An important non-trivial tool in universal algebra are ultraproducts Πi∈I Ai/U , where I is an infinite set indexing a system of σ-structures Ai, and U is an ultrafilter on I. While model theory is generally considered a part of mathematical logic, universal algebra, which grew out of Alfred North Whitehead's (1898) work on abstract algebra, is part of algebra. This is reflected by their respective MSC classifications. Nevertheless model theory can be seen as an extension of universal algebra. 23.3. FINITE MODEL THEORY 67

23.3 Finite model theory

Main article: Finite model theory

Finite model theory is the area of model theory which has the closest ties to universal algebra. Like some parts of universal algebra, and in contrast with the other areas of model theory, it is mainly concerned with finite algebras, or more generally, with finite σ-structures for signatures σ which may contain relation symbols as in the following example:

The standard signature for graphs is σᵣ={E}, where E is a binary relation symbol. A graph is a σᵣ-structure satisfying the sentences ∀u∀v(uEv → vEu) and ∀u¬(uEu) .

A σ-homomorphism is a map that commutes with the operations and preserves the relations in σ. This definition gives rise to the usual notion of graph homomorphism, which has the interesting property that a bijective homomorphism need not be invertible. Structures are also a part of universal algebra; after all, some algebraic structures such as ordered groups have a binary relation <. What distinguishes finite model theory from universal algebra is its use of more general logical sentences (as in the example above) in place of identities. (In a model-theoretic context an ′ identity t=t' is written as a sentence ∀u1u2 . . . un(t = t ) .) The logics employed in finite model theory are often substantially more expressive than first-order logic, the standard logic for model theory of infinite structures.

23.4 First-order logic

Main article: First-order logic

Whereas universal algebra provides the semantics for a signature, logic provides the . With terms, identities and quasi-identities, even universal algebra has some limited syntactic tools; first-order logic is the result of making quantification explicit and adding negation into the picture. A first-order formula is built out of atomic formulas such as R(f(x,y),z) or y = x + 1 by means of the Boolean connectives ¬, ∧, ∨, → and prefixing of quantifiers ∀v or ∃v . A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are φ (or φ(x) to mark the fact that at most x is an unbound variable in φ) and ψ defined as follows:

φ = ∀u∀v(∃w(x × w = u × v) → (∃w(x × w = u) ∨ ∃w(x × w = v))) ∧ x ≠ 0 ∧ x ≠ 1,

ψ = ∀u∀v((u × v = x) → (u = x) ∨ (v = x)) ∧ x ≠ 0 ∧ x ≠ 1. (Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the σᵣ-structure N of the natural numbers, for example, an element n satisfies the formula φ if and only if n is a . The formula ψ similarly defines irreducibility. Tarski gave a rigorous definition, sometimes called “Tarski’s definition of truth”, for the satisfaction relation |= , so that one easily proves:

N |= φ(n) ⇐⇒ n

N |= ψ(n) ⇐⇒ n A set T of sentences is called a (first-order) theory. A theory is satisfiable if it has a model M |= T , i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T. Consistency of a theory is usually defined in a syntactical way, but in first-order logic by the completeness theorem there is no need to distinguish between satisfiability and consistency. Therefore model theorists often use “consistent” as a synonym for “satisfiable”. A theory is called categorical if it determines a structure up to isomorphism, but it turns out that this definition is not useful, due to serious restrictions in the expressivity of first-order logic. The Löwenheim–Skolem theorem implies that for every theory T[2] which has an infinite model and for every infinite cardinal number κ, there is a model 68 CHAPTER 23. MODEL THEORY

M |= T such that the number of elements of M is exactly κ. Therefore only finitary structures can be described by a categorical theory. Lack of expressivity (when compared to higher logics such as second-order logic) has its advantages, though. For model theorists, the Löwenheim–Skolem theorem is an important practical tool rather than the source of Skolem’s paradox. In a certain sense made precise by Lindström’s theorem, first-order logic is the most expressive logic for which both the Löwenheim–Skolem theorem and the compactness theorem hold. As a corollary (i.e., its contrapositive), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in infinite model theory, where the words “by compactness” are commonplace. One way to prove it is by means of ultraproducts. An alternative proof uses the completeness theorem, which is otherwise reduced to a marginal role in most of modern model theory.

23.5 Axiomatizability, elimination of quantifiers, and model-completeness

The first step, often trivial, for applying the methods of model theory to a class of mathematical objects such as groups, or trees in the sense of graph theory, is to choose a signature σ and represent the objects as σ-structures. The next step is to show that the class is an elementary class, i.e. axiomatizable in first-order logic (i.e. there is a theory T such that a σ-structure is in the class if and only if it satisfies T). E.g. this step fails for the trees, since connectedness cannot be expressed in first-order logic. Axiomatizability ensures that model theory can speak about the right objects. Quantifier elimination can be seen as a condition which ensures that model theory does not say too much about the objects.

A theory T has quantifier elimination if every first-order formula φ(x1,...,xn) over its signature is equivalent modulo T to a first-order formula ψ(x1,...,xn) without quantifiers, i.e. ∀x1 ... ∀xn(ϕ(x1, . . . , xn) ↔ ψ(x1, . . . , xn)) holds in all models of T. For example the theory of algebraically closed fields in the signature σᵣᵢ=(×,+,−,0,1) has quantifier elimination because every formula is equivalent to a Boolean combination of equations between polynomials. A substructure of a σ-structure is a subset of its domain, closed under all functions in its signature σ, which is regarded as a σ-structure by restricting all functions and relations in σ to the subset. An embedding of a σ-structure A into another σ-structure B is a map f: A → B between the domains which can be written as an isomorphism of A with a substructure of B . Every embedding is an injective homomorphism, but the holds only if the signature contains no relation symbols. If a theory does not have quantifier elimination, one can add additional symbols to its signature so that it does. Early model theory spent much effort on proving axiomatizability and quantifier elimination results for specific theories, especially in algebra. But often instead of quantifier elimination a weaker property suffices: A theory T is called model-complete if every substructure of a model of T which is itself a model of T is an elementary substructure. There is a useful criterion for testing whether a substructure is an elementary substructure, called the Tarski–Vaught test. It follows from this criterion that a theory T is model-complete if and only if every first-order formula φ(x1,...,xn) over its signature is equivalent modulo T to an existential first-order formula, i.e. a formula of the following form:

∃v1 ... ∃vmψ(x1, . . . , xn, v1, . . . , vm) where ψ is quantifier free. A theory that is not model-complete may or may not have a model completion, which is a related model-complete theory that is not, in general, an extension of the original theory. A more general notion is that of model companions.

23.6 Categoricity

As observed in the section on first-order logic, first-order theories cannot be categorical, i.e. they cannot describe a unique model up to isomorphism, unless that model is finite. But two famous model-theoretic theorems deal with the weaker notion of κ-categoricity for a cardinal κ. A theory T is called κ-categorical if any two models of T that are of cardinality κ are isomorphic. It turns out that the question of κ-categoricity depends critically on whether κ is bigger than the cardinality of the language (i.e. ℵ0 + |σ|, where |σ| is the cardinality of the signature). For finite or 23.7. MODEL THEORY AND SET THEORY 69

countable signatures this means that there is a fundamental difference between ℵ0 -cardinality and κ-cardinality for uncountable κ.

A few characterizations of ℵ0 -categoricity include:

For a complete first-order theory T in a finite or countable signature the following conditions are equiv- alent:

1. T is ℵ0 -categorical. 2. For every natural number n, the Stone space Sn(T) is finite.

3. For every natural number n, the number of formulas φ(x1, ..., x) in n free variables, up to equiv- alence modulo T, is finite.

This result, due independently to Engeler, Ryll-Nardzewski and Svenonius, is sometimes referred to as the Ryll- Nardzewski theorem.

Further, ℵ0 -categorical theories and their countable models have strong ties with oligomorphic groups. They are often constructed as Fraïssé limits. Michael Morley's highly non-trivial result that (for countable languages) there is only one notion of uncountable categoricity was the starting point for modern model theory, and in particular classification theory and stability theory:

Morley’s categoricity theorem If a first-order theory T in a finite or countable signature is κ-categorical for some uncountable cardinal κ, then T is κ-categorical for all uncountable cardinals κ.

Uncountably categorical (i.e. κ-categorical for all uncountable cardinals κ) theories are from many points of view the most well-behaved theories. A theory that is both ℵ0 -categorical and uncountably categorical is called totally categorical.

23.7 Model theory and set theory

Set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem’s paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model. Particularly the proof of the independence of the continuum hypothesis requires considering sets in models which appear to be uncountable when viewed from within the model, but are countable to someone outside the model. The model-theoretic viewpoint has been useful in set theory; for example in Kurt Gödel’s work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philo- sophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory. In the other direction, model theory itself can be formalized within ZFC set theory. The development of the funda- mentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC frame- work. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension. Neither of these results are provable in ZFC alone. Finally, some questions arising from model theory (such as compactness for infinitary logics) have been shown to be equivalent to large cardinal axioms.

23.8 Other basic notions of model theory

23.8.1 Reducts and expansions

Main article: Reduct 70 CHAPTER 23. MODEL THEORY

A field or a can be regarded as a (commutative) group by simply ignoring some of its structure. The corresponding notion in model theory is that of a reduct of a structure to a subset of the original signature. The opposite relation is called an expansion - e.g. the (additive) group of the rational numbers, regarded as a structure in the signature {+,0} can be expanded to a field with the signature {×,+,1,0} or to an ordered group with the signature {+,0,<}. Similarly, if σ' is a signature that extends another signature σ, then a complete σ'-theory can be restricted to σ by intersecting the set of its sentences with the set of σ-formulas. Conversely, a complete σ-theory can be regarded as a σ'-theory, and one can extend it (in more than one way) to a complete σ'-theory. The terms reduct and expansion are sometimes applied to this relation as well.

23.8.2 Interpretability

Main article: Interpretation (model theory)

Given a mathematical structure, there are very often associated structures which can be constructed as a of part of the original structure via an . An important example is a of a group. One might say that to understand the full structure one must understand these . When the equivalence relation is definable, we can give the previous sentence a precise meaning. We say that these structures are interpretable. A key fact is that one can translate sentences from the language of the interpreted structures to the language of the original structure. Thus one can show that if a structure M interprets another whose theory is undecidable, then M itself is undecidable.

23.8.3 Using the compactness and completeness theorems

Gödel’s completeness theorem (not to be confused with his incompleteness theorems) says that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory. This is the heart of model theory as it lets us answer questions about theories by looking at models and vice versa. One should not confuse the completeness theorem with the notion of a complete theory. A complete theory is a theory that contains every sentence or its negation. Importantly, one can find a complete consistent theory extending any consistent theory. However, as shown by Gödel’s incompleteness theorems only in relatively simple cases will it be possible to have a complete consistent theory that is also recursive, i.e. that can be described by a recursively enumerable set of axioms. In particular, the theory of natural numbers has no recursive complete and consistent theory. Non-recursive theories are of little practical use, since it is undecidable if a proposed axiom is indeed an axiom, making proof-checking a supertask. The compactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. In the context of proof theory the analogous statement is trivial, since every proof can have only a finite number of antecedents used in the proof. In the context of model theory, however, this proof is somewhat more difficult. There are two well known proofs, one by Gödel (which goes via proofs) and one by Malcev (which is more direct and allows us to restrict the cardinality of the resulting model). Model theory is usually concerned with first-order logic, and many important results (such as the completeness and compactness theorems) fail in second-order logic or other alternatives. In first-order logic all infinite cardinals look the same to a language which is countable. This is expressed in the Löwenheim–Skolem theorems, which state that any countable theory with an infinite model A has models of all infinite cardinalities (at least that of the language) which agree with A on all sentences, i.e. they are 'elementarily equivalent'.

23.8.4 Types

Main article: Type (model theory)

Fix an L -structure M , and a natural number n . The set of definable subsets of M n over some parameters A is a Boolean algebra. By Stone’s representation theorem for Boolean algebras there is a natural dual notion to this. One can consider this to be the consisting of maximal consistent sets of formulae over A . We call this the space of (complete) n -types over A , and write Sn(A) . 23.9. HISTORY 71

n Now consider an element m ∈ M . Then the set of all formulae ϕ with parameters in A in free variables x1, . . . , xn so that M |= ϕ(m) is consistent and maximal such. It is called the type of m over A . One can show that for any n -type p , there exists some elementary extension N of M and some a ∈ N n so that p is the type of a over A . Many important properties in model theory can be expressed with types. Further many proofs go via constructing models with elements that contain elements with certain types and then using these elements. Illustrative Example: Suppose M is an algebraically closed field. The theory has quantifier elimination . This allows us to show that a type is determined exactly by the polynomial equations it contains. Thus the space of n -types over a subfield A is bijective with the set of prime ideals of the polynomial ring A[x1, . . . , xn] . This is the same set as the spectrum of A[x1, . . . , xn] . Note however that the topology considered on the type space is the constructible topology: a set of types is basic open iff it is of the form {p : f(x) = 0 ∈ p} or of the form {p : f(x) ≠ 0 ∈ p} . This is finer than the Zariski topology.

23.9 History

Model theory as a subject has existed since approximately the middle of the 20th century. However some earlier research, especially in mathematical logic, is often regarded as being of a model-theoretical nature in retrospect. The first significant result in what is now model theory was a special case of the downward Löwenheim–Skolem theorem, published by Leopold Löwenheim in 1915. The compactness theorem was implicit in work by ,[3] but it was first published in 1930, as a lemma in Kurt Gödel's proof of his completeness theorem. The Löwenheim– Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from Anatoly Maltsev.

23.10 See also

23.11 Notes

[1] Chang and Keisler, p. 1.

[2] In a countable signature. The theorem has a straightforward generalization to uncountable signatures.

[3] “All three commentators [i.e. Vaught, van Heijenoort and Dreben] agree that both the completeness and compactness theorems were implicit in Skolem 1923….” [Dawson, J. W. (1993). “The compactness of first-order logic:from gödel to lindström”. History and Philosophy of Logic 14: 15. doi:10.1080/01445349308837208.]

23.12 References

23.12.1 Canonical textbooks

• Chang, Chen Chung; Keisler, H. Jerome (1990) [1973]. Model Theory. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier. ISBN 978-0-444-88054-3.

• Hodges, Wilfrid (1997). A shorter model theory. Cambridge: Cambridge University Press. ISBN 978-0-521- 58713-6.

• Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. Springer. ISBN 0-387-98760-6.

23.12.2 Other textbooks

• Bell, John L.; Slomson, Alan B. (2006) [1969]. Models and Ultraproducts: An Introduction (reprint of 1974 ed.). Dover Publications. ISBN 0-486-44979-3. 72 CHAPTER 23. MODEL THEORY

• Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1994). Mathematical Logic. Springer. ISBN 0-387-94258-0. • Hinman, Peter G. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.

• Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3. • Manzano, Maria (1999). Model theory. Oxford University Press. ISBN 0-19-853851-0.

• Poizat, Bruno (2000). A Course in Model Theory. Springer. ISBN 0-387-98655-3. • Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). New York: Springer Science+Business Media. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1220-6. • Rothmaler, Philipp (2000). Introduction to Model Theory (new ed.). Taylor & Francis. ISBN 90-5699-313-5.

• Ziegler, Martin; Tent, Katrin (2012). A Course in Model Theory. Cambridge University Press. ISBN 9780521763240.

23.12.3 Free online texts

• Chatzidakis, Zoe (2001). Introduction to Model Theory (PDF). pp. 26 pages.

• Pillay, Anand (2002). Lecture Notes – Model Theory (PDF). pp. 61 pages. • Hazewinkel, Michiel, ed. (2001), “Model theory”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4

• Hodges, Wilfrid, Model theory. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.). • Hodges, Wilfrid, First-order Model theory. The Stanford Encyclopedia Of Philosophy, E. Zalta (ed.).

• Simmons, Harold (2004), An introduction to Good old fashioned model theory. Notes of an introductory course for postgraduates (with exercises).

• J. Barwise and S. Feferman (editors), Model-Theoretic Logics, Perspectives in Mathematical Logic, Volume 8, New York: Springer-Verlag, 1985. Chapter 24

Normal measure

In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower construction. Equivalently, if f:κ→κ is such that f(α)<α for most α<κ, then there is a β<κ such that f(α)=β for most α<κ. (Here, “most” means that the set of elements of κ where the property holds is a member of the ultrafilter, i.e. has measure 1.) Also equivalent, the ultrafilter (set of sets of measure 1) is closed under diagonal intersection. For a normal measure, any closed unbounded (club) subset of κ contains most ordinals less than κ. And any subset containing most ordinals less than κ is stationary in κ. If an uncountable cardinal κ has a measure on it, then it has a normal measure on it.

24.1 See also

• Measurable cardinal

• Club set

24.2 References

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (1st ed.). Springer. ISBN 3-540-57071-3. pp 52–53

73 Chapter 25

Ordinal number

This article is about the mathematical concept. For number words denoting a position in a sequence (“first”, “second”, “third”, etc.), see Ordinal number (linguistics). In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. Ordinals were introduced by Georg Cantor in 1883[1] to accommodate infinite sequences and to classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[2] Two sets S and S' have the same cardinality if there is a bijection between them (i.e. there exists a function f that is both injective and surjective, that is it maps each element x of S to a unique element y = f(x) of S' and each element y of S' comes from exactly one such element x of S). If a partial order < is defined on set S, and a partial order <' is defined on set S' , then the posets (S,<) and (S' ,<') are order isomorphic if there is a bijection f that preserves the ordering. That is, f(a) <' f(b) if and only if a < b. Every well-ordered set (S,<) is order isomorphic to the set of ordinals less than one specific ordinal number [the order type of (S,<)] under their natural ordering. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω, which is identified with the cardinal number ℵ0 . However in the transfinite case, beyond ω, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely ℵ0 itself, there are uncountably many countably infinite ordinals, namely

2 3 ω ωω ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω , …, ω , …, ω , …, ω , …, ε0, ….

Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1 and likewise, 2·ω is ω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω1, which is identified with the cardinal ℵ1 (next cardinal after ℵ0 ). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals. In general, each ordinal α is the order type of the set of ordinals strictly less than the ordinal α itself. This property permits every ordinal to be represented as the set of all ordinals less than it. Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various cofinalities). Given a class of ordinals, one can identify the α- th member of that class, i.e. one can index (count) them. Such a class is closed and unbounded if its indexing function is continuous and never stops. The Cantor normal form uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential ε0 representations as ε0 = ω . Larger and larger ordinals can be defined, but they become more and more difficult to describe. Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element.

74 25.1. ORDINALS EXTEND THE NATURAL NUMBERS 75 0

ω 1

ω·5 ω+1 ω² ω+2 ω·4 ω+3 2 ω²+1 ω²+2

ω²·4 ω³ ω+4

ω⁴ ω³+ω ω ω·3 ω²·3 ω ω³+ω² ω²+ω 3 ω³·2 ω²+ω·2 ω²·2 4

ω·2+3

ω·2+2

ω·2+1 ω·2 5

Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω

25.1 Ordinals extend the natural numbers

A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide; there is only one way to put a finite set into a linear sequence, up to isomorphism. When dealing with infinite sets one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which is generalized by the ordinal numbers described here. This is because, while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below. Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some 76 CHAPTER 25. ORDINAL NUMBER

mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way) in which there is no infinite decreasing sequence (however, there may be infinite increasing sequences); equivalently, every non-empty subset of the set has a least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, “and so on”) and to measure the “length” of the whole set by the least ordinal that is not a label for an element of the set. This “length” is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it: in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less than it, i.e., the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generally identified as the set {0,1,2,…,41}. Conversely, any set (S) of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is (or can be identified with) an ordinal. So far we have mentioned only finite ordinals, which are the natural numbers. But there are infinite ones as well: the smallest infinite ordinal is ω, which is the order type of the natural numbers (finite ordinals) and that can even be identified with the set of natural numbers (indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed it can be identified with the ordinal associated with it, which is exactly how we define ω).

A graphical “matchstick” representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers.

Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals we form in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinal associated with it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωω², and much later on ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). We can go on in this way indefinitely far (“indefinitely far” is exactly what ordinals are good at: basically every time one says “and so on” when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1. 25.2. DEFINITIONS 77

25.2 Definitions

25.2.1 Well-ordered sets

Further information: Ordered set

In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is totally ordered and there is no infinite decreasing sequence, something perhaps easier to visualize. In practice, the importance of well-ordering is justified by the possibility of applying transfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered in such a way that each step is followed by a “lower” step, then the computation will terminate. Now we don't want to distinguish between two well-ordered sets if they only differ in the “labeling of their elements”, or more formally: if we can pair off the elements of the first set with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphism and the two well-ordered sets are said to be order-isomorphic, or similar (obviously this is an equivalence relation). Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a “canonical” representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. So we essentially wish to define an ordinal as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of “being order-isomorphic”. There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. We will say that the ordinal is the order type of any set in the class.

25.2.2 Definition of an ordinal as an equivalence class

The original definition of ordinal number, found for example in Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine’s axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).

25.2.3 Von Neumann definition of ordinals

Rather than defining an ordinal as an equivalence class of well-ordered sets, we will define it as a particular well- ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number. The standard definition, suggested by , is: each ordinal is the well-ordered set of all smaller ordinals. In symbols, λ = [0,λ).[3][4] Formally:

A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and every element of S is also a subset of S.

Note that the natural numbers are ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}. It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving bijective function between them. Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals S and T, S is an element of T if and only if S is a proper subset of T. Moreover, either S is an element of T, or T is an element of S, or they are 78 CHAPTER 25. ORDINAL NUMBER equal. So every set of ordinals is totally ordered. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered. Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S. For example, every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set’s size, by the . The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its strict ordering by membership. This is the Burali-Forti paradox. The class of all ordinals is variously called “Ord”, “ON”, or "∞". An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its subsets has a maximum.

25.2.4 Other definitions

There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x:

• x is an ordinal,

• x is a transitive set, and set membership is trichotomous on x,

• x is a transitive set totally ordered by set inclusion,

• x is a transitive set of transitive sets.

These definitions cannot be used in non-well-founded set theories. In set theories with urelements, one has to further make sure that the definition excludes urelements from appearing in ordinals.

25.3 Transfinite sequence

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω.

25.4 Transfinite induction

Main article: Transfinite induction

25.4.1 What is transfinite induction?

Transfinite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restating here.

Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals.

That is, if P(α) is true whenever P(β) is true for all β<α, then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β<α. 25.4. TRANSFINITE INDUCTION 79

25.4.2 Transfinite recursion

Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by transfinite recursion – the proof that the result is well-defined uses transfinite induction. Let F denote a (class) function F to be defined on the ordinals. The idea now is that, in defining F(α) for an unspecified ordinal α, one may assume that F(β) is already defined for all β < α and thus give a formula for F(α) in terms of these F(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α. Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function F by letting F(α) be the smallest ordinal not in the set {F(β) | β < α}, that is, the set consisting of all F(β) for β < α. This definition assumes the F(β) known in the very process of defining F; this apparent vicious is exactly what definition by transfinite recursion permits. In fact, F(0) makes sense since there is no ordinal β < 0, and the set {F(β) | β < 0} is empty. So F(0) is equal to 0 (the smallest ordinal of all). Now that F(0) is known, the definition applied to F(1) makes sense (it is the smallest ordinal not in the set {F(0)} = {0}), and so on (the and so on is exactly transfinite induction). It turns out that this example is not very exciting, since provably F(α) = α for all ordinals α, which can be shown, precisely, by transfinite induction.

25.4.3 Successor and limit ordinals

Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is α ∪ {α} since its elements are those of α and α itself.[3] A nonzero ordinal that is not a successor is called a limit ordinal. One justification for this term is that a limit ordinal is indeed the limit in a topological sense of all smaller ordinals (under the order topology).

When ⟨αι|ι < γ⟩ is an ordinal-indexed sequence, indexed by a limit γ and the sequence is increasing, i.e. αι < αρ whenever ι < ρ,we define its limit to be the least upper bound of the set {αι|ι < γ},that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals. Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:

There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.

So in the following sequence:

0, 1, 2, ... , ω, ω+1

ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) we can find another ordinal (natural number) larger than it, but still less than ω. Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite induction rely upon it. Very often, when defining a function F by transfinite induction on all ordinals, one defines F(0), and F(α+1) assuming F(α) is defined, and then, for limit ordinals δ one defines F(δ) as the limit of the F(β) for all β<δ (either in the sense of ordinal limits, as we have just explained, or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) are called continuous. We will see that ordinal addition, multiplication and exponentiation are continuous as functions of their second argument.

25.4.4 Indexing classes of ordinals

We have mentioned that any well-ordered set is similar (order-isomorphic) to a unique ordinal number α , or, in other words, that its elements can be indexed in increasing fashion by the ordinals less than α . This applies, in particular, 80 CHAPTER 25. ORDINAL NUMBER

to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some α . The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So we can freely speak of the γ -th element in the class (with the convention that the “0-th” is the smallest, the “1-th” is the next smallest, and so on). Formally, the definition is by transfinite induction: the γ -th element of the class is defined (provided it has already been defined for all β < γ ), as the smallest element greater than the β -th element for all β < γ . We can apply this, for example, to the class of limit ordinals: the γ -th ordinal, which is either a limit or zero is ω · γ (see ordinal arithmetic for the definition of multiplication of ordinals). Similarly, we can consider additively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the γ -th additively indecomposable ordinal is indexed as ωγ . The technique of indexing classes of ordinals is often useful in α the context of fixed points: for example, the γ -th ordinal α such that ω = α is written εγ . These are called the "epsilon numbers".

25.4.5 Closed unbounded sets and classes

A class C of ordinals is said to be unbounded, or cofinal, when given any ordinal α , there is a β in C such that α < β (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F is continuous in the sense that, for δ a limit ordinal, F (δ) (the δ -th ordinal in the class) is the limit of all F (γ) for γ < δ ; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent). Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the termi- nology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ε· ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded. A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality. Rather than formulating these definitions for (proper) classes of ordinals, we can formulate them for sets of ordinals below a given ordinal α : A subset of a limit ordinal α is said to be unbounded (or cofinal) under α provided any ordinal less than α is less than some ordinal in the set. More generally, we can call a subset of any ordinal α cofinal in α provided every ordinal less than α is less than or equal to some ordinal in the set. The subset is said to be closed under α provided it is closed for the order topology in α , i.e. a limit of ordinals in the set is either in the set or equal to α itself.

25.5 Arithmetic of ordinals

Main article: Ordinal arithmetic

There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be de- fined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called “natural” arithmetical operations retain commutativity at the expense of continuity.

25.6 Ordinals and cardinals 25.7. SOME “LARGE” COUNTABLE ORDINALS 81

25.6.1 Initial ordinal of a cardinal

Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order-type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal. Cantor used the cardinality to partition ordinals into classes. He referred to the natural numbers as the first number class, the ordinals with cardinality ℵ0 (the countably infinite ordinals) as the second number class and generally, [5] the ordinals with cardinality ℵn−2 as the n-th number class.

The α-th infinite initial ordinal is written ωα . Its cardinality is written ℵα . For example, the cardinality of ω0 = ω 2 is ℵ0 , which is also the cardinality of ω or ε0 (all are countable ordinals). So (assuming the axiom of choice) we identify ω with ℵ0 , except that the notation ℵ0 is used when writing cardinals, and ω when writing ordinals (this is ℵ2 ℵ 2 important since, for example, 0 = 0 whereas ω > ω ). Also, ω1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and ω1 is the order type of that set), ω2 is the smallest ordinal whose cardinality is greater than ℵ1 , and so on, and ωω is the limit of the ωn for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ωn ). See also Von Neumann cardinal assignment.

25.6.2 Cofinality

The cofinality of an ordinal α is the smallest ordinal δ that is the order type of a cofinal subset of α . Notice that a number of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set. Thus for a limit ordinal, there exists a δ -indexed strictly increasing sequence with limit α . For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does ωω or an uncountable cofinality. The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least ω . An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom of Choice, then ωα+1 is regular for each α. In this case, the ordinals 0, 1, ω , ω1 , and ω2 are regular, whereas 2, 3, ωω , and ωω·₂ are initial ordinals that are not regular. The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation is idempotent.

25.7 Some “large” countable ordinals

For more details on this topic, see Large countable ordinal.

We have already mentioned (see Cantor normal form) the ordinal ε0, which is the smallest satisfying the equation ω ωα = α , so it is the limit of the sequence 0, 1, ω , ωω , ωω , etc. Many ordinals can be defined in such a manner α as fixed points of certain ordinal functions (the ι -th ordinal such that ω = α is called ει , then we could go on trying to find the ι -th ordinal such that εα = α , “and so on”, but all the subtlety lies in the “and so on”). We can try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limits CK a system of construction in this manner is the Church–Kleene ordinal, ω1 (despite the ω1 in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by a (this can CK be made rigorous, of course). Considerably large ordinals can be defined below ω1 , however, which measure the 82 CHAPTER 25. ORDINAL NUMBER

“proof-theoretic strength” of certain formal systems (for example, ε0 measures the strength of Peano arithmetic). Large ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.

25.8 Topology and ordinals

For more details on this topic, see Order topology.

Any ordinal can be made into a topological space in a natural way by endowing it with the order topology. See the Topology and ordinals section of the “Order topology” article.

25.9 Downward closed sets of ordinals

A set is downward closed if anything less than an element of the set is also in the set. If a set of ordinals is downward closed, then that set is an ordinal—the least ordinal not in the set. Examples:

• The set of ordinals less than 3 is 3 = { 0, 1, 2 }, the smallest ordinal not less than 3.

• The set of finite ordinals is infinite, the smallest infinite ordinal: ω.

• The set of countable ordinals is uncountable, the smallest uncountable ordinal: ω1.

25.10 See also

• Counting

• Ordinal space

25.11 Notes

[1] Thorough introductions are given by Levy (1979) and Jech (2003).

[2] Hallett, Michael (1979), “Towards a theory of mathematical research programmes. I”, The British Journal for the Philosophy of Science 30 (1): 1–25, doi:10.1093/bjps/30.1.1, MR 532548. See the footnote on p. 12.

[3] von Neumann 1923

[4] Levy (1979, p. 52) attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s.

[5] Dauben (1990:97)

25.12 References

• Cantor, G., (1897), Beitrage zur Begrundung der transfiniten Mengenlehre. II (tr.: Contributions to the Founding of the Theory of Transfinite Numbers II), Mathematische Annalen 49, 207-246 English translation.

• Conway, J. H. and Guy, R. K. “Cantor’s Ordinal Numbers.” In The Book of Numbers. New York: Springer- Verlag, pp. 266–267 and 274, 1996.

• Dauben, Joseph Warren, (1990), Georg Cantor: his mathematics and philosophy of the infinite. Chapter 5: The Mathematics of Cantor’s Grundlagen. ISBN 0-691-02447-2 25.13. EXTERNAL LINKS 83

• Hamilton, A. G. (1982), Numbers, Sets, and Axioms : the Apparatus of Mathematics, New York: Cambridge University Press, ISBN 0-521-24509-5 See Ch. 6, “Ordinal and cardinal numbers” • Kanamori, A., Set Theory from Cantor to Cohen, to appear in: Andrew Irvine and John H. Woods (editors), The Handbook of the Philosophy of Science, volume 4, Mathematics, Cambridge University Press. • Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag Reprinted 2002, Dover. ISBN 0-486- 42079-5 • Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag

• Sierpiński, W. (1965). Cardinal and Ordinal Numbers (2nd ed.). Warszawa: Państwowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.

• Suppes, P. (1960), Axiomatic Set Theory, D.Van Nostrand Company Inc., ISBN 0-486-61630-4 • von Neumann, Johann (1923), “Zur Einführung der trasfiniten Zahlen”, Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum 1: 199–208

• von Neumann, John (January 2002) [1923], “On the introduction of transfinite numbers”, in Jean van Hei- jenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.

25.13 External links

• Hazewinkel, Michiel, ed. (2001), “Ordinal number”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4 • Weisstein, Eric W., “Ordinal Number”, MathWorld.

• Ordinals at ProvenMath • Beitraege zur Begruendung der transfiniten Mengenlehre Cantor’s original paper published in Mathematische Annalen 49(2), 1897 • Ordinal calculator GPL'd free software for computing with ordinals and ordinal notations

• Chapter 4 of Don Monk’s lecture notes on set theory is an introduction to ordinals. Chapter 26

Partially ordered set

{x,y,z}

{x,y} {x,z} {y,z}

{x} {y} {z}

Ø

The Hasse of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal level don't share a precedence relationship. Other pairs, such as {x} and {y,z}, do not either.

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depicts the ordering relation.[1] A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

84 26.1. FORMAL DEFINITION 85

26.1 Formal definition

A (non-strict) partial order[2] is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive, i.e., which satisfies for all a, b, and c in P:

• a ≤ a (reflexivity);

• if a ≤ b and b ≤ a then a = b (antisymmetry);

• if a ≤ b and b ≤ c then a ≤ c (transitivity).

In other words, a partial order is an antisymmetric preorder. A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimes also used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as “ordered sets”, especially in areas where these structures are more common than posets. For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they are incomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partial order under which every pair of elements is comparable is called a or linear order; a totally ordered set is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no two distinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-right figure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no third element c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b.A more concise definition will be given below using the strict order corresponding to "≤". For example, {x} is covered by {x,z} in the top-right figure, but not by {x,y,z}.

26.2 Examples

Standard examples of posets arising in mathematics include:

• The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).

• The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right). Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring.

• The set of natural numbers equipped with the relation of divisibility.

• The vertex set of a directed acyclic graph ordered by reachability.

• The set of subspaces of a vector space ordered by inclusion.

• For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequence a precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn)∈ℕ if and only if a ≤ b for all n in ℕ, i.e. a componentwise order.

• For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ g if and only if f(x) ≤ g(x) for all x in X.

• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...

26.3 Extrema

There are several notions of “greatest” and “least” element in a poset P, notably: 86 CHAPTER 26. PARTIALLY ORDERED SET

• Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g. An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest or least element.

• Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a in P such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a < m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements.

• Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for each element a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is a lower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, and a least element is a lower bound of P.

For example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, which is a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not even have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset {2,3,5,10}, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound.

26.4 Orders on the Cartesian product of partially ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see figures):

• the lexicographical order:(a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);

• the product order:(a,b) ≤ (c,d) if a ≤ c and b ≤ d;

• the reflexive closure of the direct product of the corresponding strict orders: (a,b) ≤ (c,d) if (a < c and b < d) or (a = c and b = d).

All three can similarly be defined for the Cartesian product of more than two sets. Applied to ordered vector spaces over the same field, the result is in each case also an . See also orders on the Cartesian product of totally ordered sets.

26.5 Sums of partially ordered sets

Another way to combine two posets is the ordinal sum[3] (or linear sum[4]), Z = X ⊕ Y, defined on the union of the underlying sets X and Y by the order a ≤Z b if and only if:

• a, b ∈ X with a ≤X b, or

• a, b ∈ Y with a ≤Y b, or

• a ∈ X and b ∈ Y.

If two posets are well-ordered, then so is their ordinal sum.[5] 26.6. STRICT AND NON-STRICT PARTIAL ORDERS 87

26.6 Strict and non-strict partial orders

In some contexts, the partial order defined above is called a non-strict (or reflexive, or weak) partial order. In these contexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric, i.e. which satisfies for all a, b, and c in P:

• not a < a (irreflexivity), • if a < b and b < c then a < c (transitivity), and • if a < b then not b < a (asymmetry; implied by irreflexivity and transitivity[6]).

There is a 1-to-1 correspondence between all non-strict and strict partial orders. If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by:

a < b if a ≤ b and a ≠ b

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closure given by:

a ≤ b if a < b or a = b.

This is the reason for using the notation "≤". Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a

26.7 Inverse and

The inverse or converse ≥ of a partial order relation ≤ satisfies x≥y if and only if y≤x. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤. Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three. In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. The natural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitude whereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry.

26.8 Mappings between partially ordered sets

Given two partially ordered sets (S,≤) and (T,≤), a function f: S → T is called order-preserving, or monotone, or isotone, if for all x and y in S, x≤y implies f(x) ≤ f(y). If (U,≤) is also a partially ordered set, and both f: S → T and g: T → U are order-preserving, their composition (g∘f): S → U is order-preserving, too. A function f: S → T is called order-reflecting if for all x and y in S, f(x) ≤ f(y) implies x≤y. If f is both order-preserving and order-reflecting, then it is called an order-embedding of (S,≤) into (T,≤). In the latter case, f is necessarily injective, since f(x) = f(y) implies x ≤ y and y ≤ x. If an order-embedding between two posets S and T exists, one says that S can be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the 88 CHAPTER 26. PARTIALLY ORDERED SET

partial orders (S,≤) and (T,≤) are said to be isomorphic. Isomorphic orders have structurally similar Hasse (cf. right picture). It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘g yields the identity function on S and T, respectively, then S and T are order-isomorphic. [7] For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the power set of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. It is order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neither injective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Taking instead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order- reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number to the set {4}), but it can be made one by restricting its to g(ℕ). The right picture shows a subset of ℕ and its isomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called distributive lattices, see "Birkhoff’s representation theorem".

26.9 Number of partial orders

Partially ordered set of set of all subsets of a six-element set {a, b, c, d, e, f}, ordered by the subset relation.

Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements: The number of strict partial orders is the same as that of partial orders. If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (sequence A000112 in OEIS).

26.10 Linear extension

A partial order ≤* on a set X is an extension of another partial order ≤ on X provided that for all elements x and y of X, whenever x ≤ y , it is also the case that x ≤* y.A linear extension is an extension that is also a linear (i.e., total) order. Every partial order can be extended to a total order (order-extension principle).[8] In computer science, algorithms for finding linear extensions of partial orders (represented as the reachability orders of directed acyclic graphs) are called topological sorting. 26.11. IN CATEGORY THEORY 89

26.11 In category theory

Every poset (and every preorder) may be considered as a category in which every hom-set has at most one element. More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initial object, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.

26.12 Partial orders in topological spaces

Main article: Partially ordered space

If P is a partially ordered set that has also been given the structure of a topological space, then it is customary to assume that {(a, b): a ≤ b} is a closed subset of the topological product space P × P . Under this assumption partial [9] order relations are well behaved at limits in the sense that if ai → a , bi → b and ai ≤ bi for all i, then a ≤ b.

26.13 Interval

For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains at least the elements a and b. Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers is empty since there are no integers i such that 1 < i < 2. Sometimes the definitions are extended to allow a > b, in which case the interval is empty. The half-open intervals [a,b) and (a,b] are defined similarly. A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural order- ing. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1). Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a,b] = {a,b}. This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders.

26.14 See also

• antimatroid, a formalization of orderings on a set that allows more general families of orderings than posets

• causal set

• comparability graph

• complete partial order

•

• graded poset

• lattice

• ordered group

• poset topology, a kind of topological space that can be defined from any poset

• Scott continuity - continuity of a function between two partial orders.

• semilattice 90 CHAPTER 26. PARTIALLY ORDERED SET

• semiorder

• series-parallel partial order • stochastic dominance

• strict weak ordering - strict partial order "<" in which the relation “neither a < b nor b < a" is transitive. • Zorn’s lemma

26.15 Notes

[1] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons. p. 28. ISBN 0-471-83817-9. Retrieved 27 July 2012. A partially ordered set is conveniently represented by a Hasse diagram...

[2] Simovici, Dan A. & Djeraba, Chabane (2008). “Partially Ordered Sets”. Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. Springer. ISBN 9781848002012.

[3] Neggers, J.; Kim, Hee Sik (1998), “4.2 Product Order and Lexicographic Order”, Basic Posets, World Scientific, pp. 62–63, ISBN 9789810235895

[4] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 17-18

[5] P. R. Halmos (1974). Naive Set Theory. Springer. p. 82. ISBN 978-1-4757-1645-0.

[6] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as “strictly antisymmetric”.

[7] Davey, B. A.; Priestley, H. A. (2002). “Maps between ordered sets”. Introduction to Lattices and Order (2nd ed.). New York: Cambridge University Press. pp. 23–24. ISBN 0-521-78451-4. MR 1902334.

[8] Jech, Thomas (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 0-486-46624-8.

[9] Ward, L. E. Jr (1954). “Partially Ordered Topological Spaces”. Proceedings of the American Mathematical Society 5 (1): 144–161. doi:10.1090/S0002-9939-1954-0063016-5

26.16 References

• Deshpande, Jayant V. (1968). “On Continuity of a Partial Order”. Proceedings of the American Mathematical Society 19 (2): 383–386. doi:10.1090/S0002-9939-1968-0236071-7. • Schröder, Bernd S. W. (2003). Ordered Sets: An Introduction. Birkhäuser, Boston.

• Stanley, Richard P.. Enumerative Combinatorics 1. Cambridge Studies in Advanced Mathematics 49. Cam- bridge University Press. ISBN 0-521-66351-2.

26.17 External links

• A001035: Number of posets with n labeled elements in the OEIS

• A000112: Number of posets with n unlabeled elements in the OEIS Chapter 27

Power set

For the search engine developer, see Powerset (company). In mathematics, the power set (or powerset) of any set S, written P(S) , ℘(S), P(S), ℙ(S) or 2S, is the set of all

{x,y,z}

{x,y} {x,z} {y,z}

{x} {y} {z}

Ø

The elements of the power set of the set {x, y, z} ordered in respect to inclusion. subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.[1] Any subset of P(S) is called a family of sets over S.

27.1 Example

If S is the set {x, y, z}, then the subsets of S are:

91 92 CHAPTER 27. POWER SET

• {} (also denoted ∅ , the empty set) • {x} • {y} • {z} • {x, y} • {x, z} • {y, z} • {x, y, z}

and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.[2]

27.2 Properties

If S is a finite set with |S| = n elements, then the number of subsets of S is |P(S)| = 2n . This fact, which is the motivation for the notation 2S, may be demonstrated simply as follows,

We write any subset of S in the format {ω1, ω2, . . . , ωn} where ωi, 1 ≤ i ≤ n , can take the value of 0 or 1 . If ωi = 1 , the i -th element of S is in the subset; otherwise, the i -th element is not in the subset. Clearly the number of distinct subsets that can be constructed this way is 2n .

Cantor’s diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set). In particular, Cantor’s theorem shows that the power set of a countably infinite set is uncountably infinite. For example, the power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone’s representation theorem). The power set of a set S forms an abelian group when considered with the operation of symmetric difference (with the empty set as the identity element and each set being its own inverse) and a commutative monoid when considered with the operation of intersection. It can hence be shown (by proving the distributive laws) that the power set considered together with both of these operations forms a .

27.3 Representing subsets as functions

In set theory, XY is the set of all functions from Y to X. As “2” can be defined as {0,1} (see natural number), 2S (i.e., {0,1}S) is the set of all functions from S to {0,1}. By identifying a function in 2S with the corresponding preimage of 1, we see that there is a bijection between 2S and P(S) , where each function is the characteristic function of the subset in P(S) with which it is identified. Hence 2S and P(S) could be considered identical set-theoretically. (Thus there are two distinct notational motivations for denoting the power set by 2S: the fact that this function-representation of subsets makes it a special case of the XY notation and the property, mentioned above, that |2S| = 2|S|.) This notion can be applied to the example above in which S = {x, y, z} to see the isomorphism with the binary numbers from 0 to 2n−1 with n being the number of elements in the set. In S, a 1 in the position corresponding to the location in the set indicates the presence of the element. So {x, y} = 110. For the whole power set of S we get:

• { } = 000 (Binary) = 0 (Decimal) 27.4. RELATION TO BINOMIAL THEOREM 93

• {x} = 100 = 4

• {y} = 010 = 2

• {z} = 001 = 1

• {x, y} = 110 = 6

• {x, z} = 101 = 5

• {y, z} = 011 = 3

• {x, y, z} = 111 = 7

27.4 Relation to binomial theorem

The power set is closely related to the binomial theorem. The number of sets with k elements in the power set of a set with n elements will be a combination C(n, k), also called a binomial coefficient. For example the power set of a set with three elements, has:

• C(3, 0) = 1 set with 0 elements

• C(3, 1) = 3 sets with 1 element

• C(3, 2) = 3 sets with 2 elements

• C(3, 3) = 1 set with 3 elements.

27.5 Algorithms

If S is a finite set, there is a recursive algorithm to calculate P(S) . Define the operation F(e, T ) = {X ∪ {e}|X ∈ T } In English, return the set with the element eadded to each set X in T .

• If S = {},then P(S) = {{}} is returned.

• Otherwise:

• Let ebe any single element of S . • Let T = S \{e}, where ' S \{e}' denotes the relative complement of {e}in S . • And the result: P(S) = P(T ) ∪ F(e, P(T )) is returned.

In other words, the power set of the empty set is the set containing the empty set and the power set of any other set is all the subsets of the set containing some specific element and all the subsets of the set not containing that specific element.

27.6 Subsets of limited cardinality

The set of subsets of S of cardinality less than κ is denoted by Pκ(S) or P<κ(S) . Similarly, the set of non-empty subsets of S might be denoted by P≥1(S) . 94 CHAPTER 27. POWER SET

27.7 Power object

A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra. Now the power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard subalgebras behave analogously to subsets. However there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way. Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs G and H, a homomorphism h: G → H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set HG of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of G as the multigraph ΩG, called the power object of G. What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s,t: E → V giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a presheaf. Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Ω, called a subobject classifier. Although the term “power object” is sometimes used synonymously with exponential object YX, in topos theory Y is required to be Ω.

27.8 Functors and quantifiers

In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint.[3]

27.9 See also

• Set theory • Axiomatic set theory • Family of sets • Field of sets

27.10 Notes

[1] Devlin (1979) p.50

[2] Puntambekar, A.A. (2007). Theory Of Automata And Formal Languages. Technical Publications. pp. 1–2. ISBN 978- 81-8431-193-8. 27.11. REFERENCES 95

[3] , Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 See page 58

27.11 References

• Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN 0-387-90441-7. Zbl 0407.04003.

• Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403.

• Puntambekar, A.A. (2007). Theory Of Automata And Formal Languages. Technical Publications. ISBN 978-81-8431-193-8.

27.12 External links

• Weisstein, Eric W., “Power Set”, MathWorld.

• Power set at PlanetMath.org. • Power set in nLab

• Power object in nLab Chapter 28

Preorder

Not to be confused with Pre-order. This article is about binary relations. For the graph vertex ordering, see Depth-first search. For other uses, see Preorder (disambiguation). “Quasiorder” redirects here. For irreflexive transitive relations, see strict order.

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. All partial orders and equivalence relations are preorders, but preorders are more general. The name 'preorder' comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they're neither necessarily anti-symmetric nor symmetric. Because a preorder is a binary relation, the symbol ≤ can be used as the notational device for the relation. However, because they are not necessarily anti- symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied. In words, when a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or ≲ is used instead of ≤. To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. Note that, in general, the corresponding graphs may be cyclic graphs: preorders may have cycles in them. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder may have many disconnected components.

28.1 Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity) if a ≤ b and b ≤ c then a ≤ c (transitivity)

A set that is equipped with a preorder is called a preordered set (or proset).[1] If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order. On the other hand, if it is symmetric, that is, if a ≤ b implies b ≤ a, then it is an equivalence relation. A preorder which is preserved in all contexts (i.e. respected by all functions on P) is called a precongruence.A precongruence which is also symmetric (i.e. is an equivalence relation) is a congruence relation.

96 28.2. EXAMPLES 97

Equivalently, a preordered set P can be defined as a category with objects the elements of P, and each hom-set having at most one element (one for objects which are related, zero otherwise). Alternately, a preordered set can be understood as an enriched category, enriched over the category 2 = (0→1). A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class. Preordered classes can be defined as thin categories, i.e. as categories with at most one from an object to another.

28.2 Examples

• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where x ≤ y in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for every pair (x, y) with x ≤ y). However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partially ordered sets (preorders satisfying an additional anti-symmetry property).

• Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way. That is, there is a 1-to-1 correspondence between finite and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not 1-to-1.

• A is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without losing important features.

• The relation defined by x ≤ y if f(x) ≤ f(y) , where f is a function into some preorder.

• The relation defined by x ≤ y if there exists some injection from x to y. Injection may be replaced by surjection, or any type of structure-preserving function, such as ring homomorphism, or permutation.

• The embedding relation for countable total orderings.

• The graph-minor relation in graph theory.

• A category with at most one morphism from any object x to any other object y is a preorder. Such categories are called thin. In this sense, categories “generalize” preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.

In computer science, one can find examples of the following preorders.

• Many-one and Turing reductions are preorders on complexity classes.

• The subtyping relations are usually preorders.

• Simulation preorders are preorders (hence the name).

• Reduction relations in abstract rewriting systems.

• The encompassment preorder on the set of terms, defined by s≤t if a subterm of t is a substitution instance of s.

Example of a total preorder:

• Preference, according to common models. 98 CHAPTER 28. PREORDER

28.3 Uses

Preorders play a pivotal role in several situations:

• Every preorder can be given a topology, the Alexandrov topology; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

• Preorders may be used to define interior algebras.

• Preorders provide the Kripke semantics for certain types of modal logic.

28.4 Constructions

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure,R+=. The transitive closure indicates path connection in R: x R+ y if and only if there is an R-path from x to y. Given a preorder ≲ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a ≲ b and b ≲ a. (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition.) Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all equivalence classes of ~. Note that if the preorder is R+=, S / ~ is the set of R-cycle equivalence classes: x ∈ [y] if and only if x = y or x is in an R-cycle with y. In any case, on S / ~ we can define [x] ≤ [y] if and only if x ≲ y. By the construction of ~, this definition is independent of the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set. Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 corre- spondence between preorders and pairs (partition, partial order). For a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a ≲ b and not b ≲ a), or equivalently, using the equivalence relation introduced above, (a ≲ b and not a ~ b). It is a strict partial order; every strict partial order can be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b). (Alternatively, for a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a ≲ b and a ≠ b). The result is the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive, and if it is, as we have seen, it is the same as before.) Conversely we have a ≲ b if and only if a < b or a ~ b. This is the reason for using the notation " ≲ "; "≤" can be confusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b. Note that with this construction multiple preorders " ≲ " can give the same relation "<", so without more information, such as the equivalence relation, " ≲ " cannot be reconstructed from "<". Possible preorders include the following:

• Define a ≤ b as a < b or a = b (i.e., take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, so we don't need the notations ≲ and ~.

• Define a ≲ b as “not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a ~ b as “neither a < b nor b < a"; these relations ≲ and ~ are in general not transitive; however, if they are, ~ is an equivalence; in that case "<" is a strict weak order. The resulting preorder is total, that is, a total preorder.

28.5 Number of preorders

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:

• for n=3: 28.6. INTERVAL 99

• 1 partition of 3, giving 1 preorder • 3 partitions of 2+1, giving 3 × 3 = 9 preorders • 1 partition of 1+1+1, giving 19 preorders

i.e. together 29 preorders.

• for n=4:

• 1 partition of 4, giving 1 preorder • 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders • 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders • 1 partition of 1+1+1+1, giving 219 preorders

i.e. together 355 preorders.

28.6 Interval

For a ≲ b, the interval [a,b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b. It contains at least the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty. Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b. Also [a,b) and (a,b] can be defined similarly.

28.7 See also

• partial order - preorder that is antisymmetric • equivalence relation - preorder that is symmetric

• total preorder - preorder that is total

• total order - preorder that is antisymmetric and total • directed set

• category of preordered sets • prewellordering

• Well-quasi-ordering

28.8 References

[1] For “proset”, see e.g. Eklund, Patrik; Gähler, Werner (1990), “Generalized Cauchy spaces”, Mathematische Nachrichten 147: 219–233, doi:10.1002/mana.19901470123, MR 1127325.

• Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9 Chapter 29

Ramsey cardinal

In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962) and named after Frank P. Ramsey. With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number κ such that for every function

f: [κ]<ω → {0, 1} there is a set A of cardinality κ that is homogeneous for f (i.e.: for every n, f is constant on the subsets of cardinality n from A) is called Ramsey. A cardinal κ is called almost Ramsey if for every function

f: [κ]<ω → {0, 1} and for every λ < κ, there is a set of order type λ that is homogeneous for f. The existence of a Ramsey cardinal is sufficient to prove the existence of 0#. In fact, if κ is Ramsey, then every set with rank less than κ has a sharp. Every measurable cardinal is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal. A property intermediate in strength between Ramseyness and measurability is existence of a κ-complete normal non-principal ideal I on κ such that for every A ∉ I and for every function

f: [κ]<ω → {0, 1} there is a set B ⊂ A not in I that is homogeneous for f. If I is taken to be the ideal of nonstationary sets, this property defines the ineffably Ramsey cardinals. The existence of Ramsey cardinal implies that the existence of the cardinal and this in turn implies the falsity of Axiom of Constructibility of Kurt Gödel.

29.1 References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.

• Erdős, Paul; Hajnal, András (1962), “Some remarks concerning our paper “On the structure of set-mappings. Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal”, Acta Mathematica Academiae Scientiarum Hungaricae 13: 223–226, doi:10.1007/BF02033641, ISSN 0001-5954, MR 0141603 • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.

100 Chapter 30

Rank-into-rank

In set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength):

• Axiom I3: There is a nontrivial elementary embedding of Vλ into itself. • Axiom I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ where λ is the first fixed point above the critical point. • Axiom I1: There is a nontrivial elementary embedding of Vλ₊₁ into itself. • Axiom I0: There is a nontrivial elementary embedding of L(Vλ₊₁) into itself with the critical point below λ.

These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom for Reinhardt cardinals is stronger, but is not consistent with the axiom of choice. If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit of jn(κ) as n goes to ω. More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementary embedding of Vα into itself then α is either a limit ordinal of cofinality ω or the successor of such an ordinal. The axioms I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen’s inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent. Every I0 cardinal κ (speaking here of the critical point of j) is an I1 cardinal. Every I1 cardinal κ is an I2 cardinal and has a stationary set of I2 cardinals below it. Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it. Every I3 cardinal κ has another I3 cardinal above it and is an n-huge cardinal for every n<ω. Axiom I1 implies that Vλ₊₁ (equivalently, H(λ+)) does not satisfy V=HOD. There is no set S⊂λ definable in Vλ₊₁ (even from parameters Vλ and ordinals <λ+) with S cofinal in λ and |S|<λ, that is, no such S witnesses that λ is singular. And similarly for Axiom I0 and ordinal definability in L(Vλ₊₁) (even from parameters in Vλ). However globally, and even in Vλ,[1] V=HOD is relatively consistent with Axiom I1.

30.1 References

• Gaifman, Haim (1974), “Elementary embeddings of models of set-theory and certain subtheories”, Axiomatic set theory, Proc. Sympos. Pure Math., XIII, Part II, Providence R.I.: Amer. Math. Soc., pp. 33–101, MR 0376347 • Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3 • Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978), “Strong axioms of infinity and ele- mentary embeddings”, Annals of Mathematical Logic 13 (1): 73–116, doi:10.1016/0003-4843(78)90031-1

101 102 CHAPTER 30. RANK-INTO-RANK

[1] Consistency of V = HOD With the Wholeness Axiom, Paul Corazza, Archive for Mathematical Logic, No. 39, 2000. Chapter 31

Regular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one that cannot be broken into a smaller collection of smaller parts. If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal κ is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than κ , the elements of which are cardinals less than κ . (The situation is slightly more complicated in contexts where the axiom of choice might fail; in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above definition is restricted to well-orderable cardinals only.) An infinite ordinal α is regular if and only if it is a limit ordinal which is not the limit of a set of smaller ordinals which set has order type less than α . A regular ordinal is always an initial ordinal, though some initial ordinals are not regular. Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.

31.1 Examples

The ordinals less than ω are finite. A finite sequence of finite ordinals always has a finite maximum, so ω cannot be the limit of any sequence of type less than ω whose elements are ordinals less than ω , and is therefore a regular ordinal. ℵ0 (aleph-null) is a regular cardinal because its initial ordinal, ω , is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite. ω + 1 is the next ordinal number greater than ω . It is singular, since it is not a limit ordinal. ω + ω is the next limit ordinal after ω . It can be written as the limit of the sequence ω , ω + 1 , ω + 2 , ω + 3 , and so on. This sequence has order type ω , so ω + ω is the limit of a sequence of type less than ω + ω whose elements are ordinals less than ω + ω , therefore it is singular.

ℵ1 is the next cardinal number greater than ℵ0 , so the cardinals less than ℵ1 are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So ℵ1 cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

ℵω is the next cardinal number after the sequence ℵ0 , ℵ1 , ℵ2 , ℵ3 , and so on. Its initial ordinal ωω is the limit of the sequence ω , ω1 , ω2 , ω3 , and so on, which has order type ω , so ωω is singular, and so is ℵω . Assuming the axiom of choice, ℵω is the first infinite cardinal which is singular (the first infinite ordinal which is singular is ω + 1 ). Proving the existence of singular cardinals requires the axiom of replacement, and in fact the inability to prove the existence of ℵω in is what led Fraenkel to postulate this axiom.

31.2 Properties

Uncountable limit cardinals that are also regular are known as weakly inaccessible cardinals. They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom. Inaccessible cardinals are necessarily fixed points of the aleph function, though not all

103 104 CHAPTER 31. REGULAR CARDINAL

fixed points are regular. For instance, the first fixed point is the limit of the ω -sequence ℵ , ℵℵ , ℵℵ , ... and is 0 0 ℵ0 therefore singular. If the axiom of choice holds, then every successor cardinal is regular. Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal. Some cardinal numbers cannot be proven to be equal to any particular aleph, for instance the cardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see Easton’s theorem). The continuum hypothesis postulates that the cardinality of the continuum is equal to ℵ1 which is regular. Without the axiom of choice, there would be cardinal numbers which were not well-orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore only the aleph numbers can meaningfully be called regular or singular cardinals. Furthermore, a successor aleph need not be regular. For instance, the union of a countable set of countable sets need not be countable. It is consistent with ZF that ω1 be the limit of a countable sequence of countable ordinals as well as the set of real numbers is countable union of countable sets. Furthermore, it is consistent with ZF that every aleph bigger than ℵ0 is singular (a result proved by Moti Gitik).

31.3 See also

• Inaccessible cardinal

31.4 References

• Herbert B. Enderton, Elements of Set Theory, ISBN 0-12-238440-7

• Kenneth Kunen, Set Theory, An Introduction to Independence Proofs, ISBN 0-444-85401-0 Chapter 32

Scott’s trick

In set theory, Scott’s trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65). The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for ordinal numbers in Zermelo–Fraenkel set theory (Forster 2003:182). The method was introduced by Dana Scott (1955). Beyond the problem of defining set representatives for ordinal numbers, Scott’s trick can be used to obtain represen- tatives for cardinal numbers and more generally for isomorphism types, for example, order types of linearly ordered sets (Jech 2003:65). It is credited to be indispensable (even in the presence of the axiom of choice) when taking ultrapowers of proper classes in model theory. (Kanamori 1994:47)

32.1 Application to cardinalities

The use of Scott’s trick for cardinal numbers shows how the method is typically employed. The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them. The difficulty is that every equivalence class of this relation is a proper class, and so the equivalence classes themselves cannot be directly manipulated in set theories, such as Zermelo–Fraenkel set theory, that only deal with sets. It is often desirable in the context of set theory to have sets that are representatives for the equivalence classes. These sets are then taken to “be” cardinal numbers, by definition. In Zermelo–Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, it is possible that some sets do not have the same cardinality as any ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representative. Scott’s trick assigns representatives differently, using the fact that for every set A there is a least rank γA in the cumulative hierarchy when some set of the same cardinality as A appears. Thus one may define the representative of the cardinal number of A to be the set of all sets of rank γA that have the same cardinality as A. This definition assigns a representative to every cardinal number even when not every set can be well-ordered (an assumption equivalent to the axiom of choice). It can be carried out in Zermelo–Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.

32.2 References

• Thomas Forster (2003), Logic, Induction and Sets, Cambridge University Press. ISBN 0-521-53361-9 • Thomas Jech, Set Theory, 3rd millennium (revised) ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2 • Akihiro Kanamori: The Higher Infinite. Large Cardinals in Set Theory from their Beginnings., Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. • Scott, Dana (1955), “Definitions by abstraction in axiomatic set theory” (PDF), Bulletin of the American Math- ematical Society 61 (5): 442

105 Chapter 33

Sequence

“Sequential” redirects here. For the manual transmission, see Sequential manual transmission. For other uses, see Sequence (disambiguation).

In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them into computer memory; infinite sequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. It is, however, bounded.

106 33.1. EXAMPLES AND NOTATION 107

33.1 Examples and notation

A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list the elements. For example, the first four odd numbers form the sequence (1,3,5,7). This notation can be used for infinite sequences as well. For instance, the infinite sequence of positive odd integers can be written (1,3,5,7,...). Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples.

33.1.1 Important examples

2 3 1 1 8 5

A tiling with squares whose sides are successive Fibonacci numbers in length.

There are many important integer sequences. The prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). The study of prime numbers has important applications for mathematics and specifically number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...). Other interesting sequences include the ban numbers, whose spellings do not contain a certain letter of the alphabet. For instance, the eban numbers (do not contain 'e') form the sequence (2,4,6,30,32,34,36,40,42,...). Another sequence based on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...). For a list of important examples of integers sequences see On-line Encyclopedia of Integer Sequences. Other important examples of sequences include ones made up of rational numbers, real numbers, and complex num- bers. The sequence (.9,.99,.999,.9999,...) approaches the number 1. In fact, every real number can be written as the limit of a sequence of rational numbers. It is this fact that allows us to write any real number as the limit of a sequence of decimals. For instance, π is the limit of the sequence (3,3.1,3.14,3.141,3.1415,...). The sequence for π, however, does not have any pattern that is easily discernible by eye, unlike the sequence (0.9,0.99,...). 108 CHAPTER 33. SEQUENCE

33.1.2 Indexing

Other notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not have a pattern such as the digits of π. This section focuses on the notations used for sequences that are a map from a subset of the natural numbers. For generalizations to other countable index sets see the following section and below. The terms of a sequence are commonly denoted by a single variable, say an, where the index n indicates the nth element of the sequence.

a1 ↔ element 1st

a2 ↔ element 2nd

a3 ↔ element 3rd . . . . an−1 ↔ element (n-1)th

an ↔ element nth an+1 ↔ element (n+1)th . . . .

Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whose elements are related to the index n (the element’s position) in a simple way. For instance, the sequence of the first 10 square numbers could be written as

2 (a1, a2, ..., a10), ak = k .

This represents the sequence (1,4,9,...100). This notation is often simplified further as

10 2 (ak)k=1, ak = k .

Here the subscript {k=1} and superscript 10 together tell us that the elements of this sequence are the ak such that k = 1, 2, ..., 10. Sequences can be indexed beginning and ending from any integer. The infinity symbol ∞ is often used as the super- script to indicate the sequence including all integer k-values starting with a certain one. The sequence of all positive squares is then denoted

∞ 2 (ak)k=1, ak = k .

In cases where the set of indexing numbers is understood, such as in analysis, the subscripts and superscripts are often left off. That is, one simply writes ak for an arbitrary sequence. In analysis, k would be understood to run from 1 to ∞. However, sequences are often indexed starting from zero, as in

∞ (ak)k=0 = (a0, a1, a2, ...).

In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways.

• (1, 9, 25, ...)

2 • (a1, a3, a5, ...), ak = k • ∞ 2 (a2k−1)k=1, ak = k 33.2. FORMAL DEFINITION AND BASIC PROPERTIES 109

• ∞ − 2 (ak)k=1, ak = (2k 1) • − 2 ∞ ((2k 1) )k=1

Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations if the indexing set was understood to be the natural numbers. Finally, sequences can more generally be denoted by writing a set inclusion in the subscript, such as in

(ak)k∈N

The set of values that the index can take on is called the index set. In general, the ordering of the elements ak is specified by the order of the elements in the indexing set. When N is the index set, the element ak+1 comes after the element ak since in N, the element (k+1) comes directly after the element k.

33.1.3 Specifying a sequence by recursion

Sequences whose elements are related to the previous elements in a straightforward way are often specified using recursion. This is in contrast to the specification of sequence elements in terms of their position. To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones before it. In addition, enough initial elements must be specified so that new elements of the sequence can be specified by the rule. The principle of mathematical induction can be used to prove that a sequence is well-defined, which is to say that that every element of the sequence is specified at least once and has a single, unambiguous value. Induction can also be used to prove properties about a sequence, especially for sequences whose most natural specification is by recursion. The Fibonacci sequence can be defined using a recursive rule along with two initial elements. The rule is that each element is the sum of the previous two elements, and the first two elements are 0 and 1.

an = an−1 + an−2 , with a0 = 0 and a1 = 1.

The first ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence that is defined recursively is Recaman’s sequence, considered at the beginning of this section. We can define Recaman’s sequence by

a0 = 0 and an = an−1−n if the result is positive and not already in the list. Otherwise, an = an−1+n .

Not all sequences can be specified by a rule in the form of an equation, recursive or not, and some can be quite complicated. For example, the sequence of prime numbers is the set of prime numbers in their natural order. This gives the sequence (2,3,5,7,11,13,17,...). One can also notice that the next element of a sequence is a function of the element before, and so we can write the next element as : an+1 = f(an) This functional notation can prove useful when one wants to prove the global monotony of the sequence.

33.2 Formal definition and basic properties

There are many different notions of sequences in mathematics, some of which (e.g., ) are not covered by the definitions and notations introduced below.

33.2.1 Formal definition

A sequence is usually defined as a function whose domain is a countable totally ordered set, although in many disci- plines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset 110 CHAPTER 33. SEQUENCE

of the natural numbers to the real numbers.[1] In other words, a sequence is a map f(n): N → R. To recover our earlier notation we might identify an = f(n) for all n or just write an : N → R. In complex analysis, sequences are defined as maps from the natural numbers to the complex numbers (C).[2] In topology, sequences are often defined as functions from a subset of the natural numbers to a topological space.[3] Sequences are an important concept for studying functions and, in topology, topological spaces. An important gener- alization of sequences, called a net, is to functions from a (possibly uncountable) directed set to a topological space.

33.2.2 Finite and infinite

The length of a sequence is defined as the number of terms in the sequence. A sequence of a finite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no elements. Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element, it is called a singly infinite, or one-sided (infinite) sequence, when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence. A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( …, −4, −2, ∞ 0, 2, 4, 6, 8… ), is bi-infinite. This sequence could be denoted (2n)n=−∞ . One can interpret singly infinite sequences as elements of the semigroup ring of the natural numbers R[N], and doubly infinite sequences as elements of the group ring of the integers R[Z]. This perspective is used in the Cauchy product of sequences.

33.2.3 Increasing and decreasing

A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For a ∞ sequence (an)n=1 this can be written as an ≤ an₊₁ for all n ∈ N. If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonically decreasing if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a . The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively.

33.2.4 Bounded

If the sequence of real numbers (an) is such that all the terms, after a certain one, are less than some real number M, then the sequence is said to be bounded from above. In less words, this means an ≤ M for all n greater than N for some pair M and N. Any such M is called an upper bound. Likewise, if, for some real m, an ≥ m for all n greater than some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence is both bounded from above and bounded from below then the sequence is said to be bounded.

33.2.5 Other types of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2,4,6,...) is a subsequence of the positive integers (1,2,3,...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved. Some other types of sequences that are easy to define include:

• An integer sequence is a sequence whose terms are integers.

• A polynomial sequence is a sequence whose terms are polynomials. 33.3. LIMITS AND CONVERGENCE 111

• A positive integer sequence is sometimes called multiplicative if anm = an am for all pairs n,m such that n and [4] m are coprime. In other instances, sequences are often called multiplicative if an = na1 for all n. Moreover, the multiplicative Fibonacci sequence satisfies the recursion relation an = an₋₁ an₋₂.

33.3 Limits and convergence

Main article: Limit of a sequence One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. 1.0

0.8

0.6

0.4 n + 1 2 0.2 2n

0.0 5 10 15 20 25

The plot of a convergent sequence (a) is shown in blue. Visually we can see that the sequence is converging to the limit zero as n increases.

Continuing informally, a (singly infinite) sequence has a limit if it approaches some value L, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the value of an approaches L as n → ∞, denoted

lim an = L. n→∞ More precisely, the sequence converges if there exists a limit L such that the remaining a's are arbitrarily close to L for some n large enough. If a sequence converges to some limit, then it is convergent; otherwise it is divergent. If an gets arbitrarily large as n → ∞ we write

lim an = ∞. n→∞ 112 CHAPTER 33. SEQUENCE

In this case we say that the sequence (an) diverges, or that it converges to infinity. If an becomes arbitrarily “small” negative numbers (large in magnitude) as n → ∞ we write

lim an = −∞ n→∞ and say that the sequence diverges or converges to minus infinity.

33.3.1 Definition of convergence

∞ For sequences that can be written as (an)n=1 with an ∈ R we can write (an) with the indexing set understood as N. These sequences are most common in real analysis. The generalizations to other types of sequences are considered in the following section and the main page Limit of a sequence. Let (an) be a sequence. In words, the sequence (an) is said to converge if there exists a number L such that no matter how close we want the an to be to L (say ε-close where ε > 0), we can find a natural number N such that all terms (aN+1, aN+2, ...) are further closer to L (within ε of L). [1] This is often written more compactly using symbols. For instance,

for all ε > 0, there exists a natural number N such that L−ε < an < L+ε for all n ≥ N.

In even more compact notation

∀ϵ > 0, ∃N ∈ N s.t. ∀n ≥ N, |an − L| < ϵ.

The difference in the definitions of convergence for (one-sided) sequences in complex√ analysis and metric spaces is that the absolute value |an − L| is interpreted as the distance in the complex plane ( z∗z ), and the distance under the appropriate metric, respectively.

33.3.2 Applications and important results

Important results for convergence and limits of (one-sided) sequences of real numbers include the following. These equalities are all true at least when both sides exist. For a discussion of when the existence of the limit on one side implies the existence of the other see a real analysis text such as can be found in the references.[1][5]

• The limit of a sequence is unique.

• limn→∞(an  bn) = limn→∞ an  limn→∞ bn

• limn→∞ can = c limn→∞ an

• limn→∞(anbn) = (limn→∞ an)(limn→∞ bn)

an limn→∞ an • limn→∞ = provided limn→∞ bn ≠ 0 bn limn→∞ bn • p p limn→∞ an = [limn→∞ an]

• If an ≤ bn for all n greater than some N, then limn→∞ an ≤ limn→∞ bn .

• (Squeeze Theorem) If an ≤ cn ≤ bn for all n > N, and limn→∞ an = limn→∞ bn = L , then limn→∞ cn = L .

• If a sequence is bounded and monotonic then it is convergent.

• A sequence is convergent if and only if every subsequence is convergent. 33.4. SERIES 113

The plot of a (X), shown in blue, as X versus n. Visually, we see that the sequence appears to be converging to the limit zero as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence converges to some limit.

33.3.3 Cauchy sequences

Main article: Cauchy sequence A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is Cauchy characterization of convergence for sequences:

In the real numbers, a sequence is convergent if and only if it is Cauchy.

In contrast, in the rational numbers, e.g. the sequence defined by x1 = 1 and xn₊₁ = xn + 2/xn/2 is Cauchy, but has no rational limit, cf. here.

33.4 Series

Main article: Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence. That is, adding the first N terms of a (one-sided) sequence forms the Nth term of another sequence, called a series. Thus the N series of the sequence (a) results in another sequence (SN) given by:

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3 . . . .

SN = a1 + a2 + a3 + ··· . . . .

We can also write the nth term of the series as 114 CHAPTER 33. SEQUENCE

∑N SN = an. n=1 Then the concepts used to talk about sequences, such as convergence, carry over to series (the sequence of partial sums) and the properties can be characterized as properties of the underlying sequences (such as (an) in the last example). The limit, if it exists, of an infinite series (the series created from an infinite sequence) is written as

∑∞ lim SN = an. N→∞ n=1

33.5 Use in other fields of mathematics

33.5.1 Topology

Sequence play an important role in topology, especially in the study of metric spaces. For instance:

• A metric space is compact exactly when it is sequentially compact.

• A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences.

• A metric space is a connected space if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.

• A topological space is separable exactly when there is a dense sequence of points.

Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems to spaces without metrics.

Product topology

A product space of a sequence of topological spaces is the cartesian product of the spaces equipped with a natural topology called the product topology.

More formally, given a sequence of spaces {Xi} , define X such that

∏ X := Xi, i∈I

is the set of sequences {xi} where each xi is an element of Xi . Let the canonical projections be written as pi : X → Xi. Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.

33.5.2 Analysis

In analysis, when talking about sequences, one will generally consider sequences of the form

(x1, x2, x3,... ) or (x0, x1, x2,... )

which is to say, infinite sequences of elements indexed by natural numbers. 33.5. USE IN OTHER FIELDS OF MATHEMATICS 115

It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by xn = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N. The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

Sequence spaces

Main article: Sequence space

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a . The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.

33.5.3 Linear algebra

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences (where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

33.5.4 Abstract algebra

Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or rings.

Free monoid

Main article: Free monoid

If A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the finite sequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroup A+ is the subsemigroup of A* containing all elements except the empty sequence.

Exact sequences

Main article: Exact sequence

In the context of , a sequence

f1 f2 f3 fn G0 −→ G1 −→ G2 −→ · · · −→ Gn

of groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to the of the next: 116 CHAPTER 33. SEQUENCE

im(fk) = ker(fk+1)

Note that the sequence of groups and homomorphisms may be either finite or infinite. A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms.

Spectral sequences

Main article: Spectral sequence

In and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

33.5.5 Set theory

An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinary sequence.

33.5.6 Computing

Automata or finite state machines can typically be thought of as directed graphs, with edges labeled using some specific alphabet, Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word). The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter.

33.5.7 Streams

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical com- puter science. They are often referred to simply as sequences or streams, as opposed to finite strings. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0, 1}∞ of all infinite, binary sequences is sometimes called the Cantor space. An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to 1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalization method for proofs.[6]

33.6 Types

• ±1-sequence

• Arithmetic progression

• Cauchy sequence

• Farey sequence

• Fibonacci sequence 33.7. RELATED CONCEPTS 117

• Geometric progression

• Look-and-say sequence

• Thue–Morse sequence

33.7 Related concepts

• List (computing)

• Ordinal-indexed sequence

• Recursion (computer science)

• Tuple

• Set theory

33.8 Operations

• Cauchy product

• Limit of a sequence

33.9 See also

•

• Net (topology) (a generalization of sequences)

• On-Line Encyclopedia of Integer Sequences

• Permutation

• Recurrence relation

• Sequence space

• Set (mathematics)

33.10 References

[1] Gaughan, Edward. “1.1 Sequences and Convergence”. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.

[2] Edward B. Saff & Arthur David Snider (2003). “Chapter 2.1”. Fundamentals of Complex Analysis. ISBN 01-390-7874-6.

[3] James R. Munkres. “Chapters 1&2”. Topology. ISBN 01-318-1629-2.

[4] Lando, Sergei K. “7.4 Multiplicative sequences”. Lectures on generating functions. AMS. ISBN 0-8218-3481-9.

[5] Dawikins, Paul. “Series and Sequences”. Paul’s Online Math Notes/Calc II (notes). Retrieved 18 December 2012.

[6] Oflazer, Kemal. “FORMAL LANGUAGES, AUTOMATA AND COMPUTATION: DECIDABILITY” (PDF). cmu.edu. Carnegie-Mellon University. Retrieved 24 April 2015. 118 CHAPTER 33. SEQUENCE

33.11 External links

• The dictionary definition of sequence at Wiktionary

• Hazewinkel, Michiel, ed. (2001), “Sequence”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4

• The On-Line Encyclopedia of Integer Sequences

• Journal of Integer Sequences (free) • Sequence at PlanetMath.org. Chapter 34

Set theory

This article is about the branch of mathematics. For musical set theory, see Set theory (music). Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Al-

A A∩B B

A Venn diagram illustrating the intersection of two sets.

though any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor and in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo– Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

119 120 CHAPTER 34. SET THEORY

34.1 History

Georg Cantor

Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, how- 34.2. BASIC CONCEPTS AND NOTATION 121

ever, was founded by a single paper in 1874 by Georg Cantor: “On a Characteristic Property of All Real Algebraic Numbers”.[1][2] Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathemati- cians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century.[3] Modern understanding of infinity began in 1867–71, with Cantor’s work on number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor’s thinking and culminated in Cantor’s 1874 paper. Cantor’s work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the “infinity of infinities” ("Cantor’s paradise") resulting from the power set operation. This utility of set theory led to the article “Mengenlehre” contributed in 1898 by Arthur Schoenflies to Klein’s encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. and Ernst Zermelo independently found the simplest and best known paradox, now called Russell’s paradox: consider “the set of all sets that are not members of themselves”, which leads to a contradiction since it must be a member of itself, and not a member of itself. In 1899 Cantor had himself posed the question “What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers were treated to Theory of Sets of Points[4] by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas category theory is thought to be a preferred foundation.

34.2 Basic concepts and notation

Main articles: Set (mathematics) and Algebra of sets

Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , and so is {2} but is not. From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule this out, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but B is not a subset of A. Note also that 1 and 2 and 3 are members (elements) of set {1,2,3} , but are not subsets, and the subsets in turn are not as such members of the set. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:

• Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .

• Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .

• Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A. The set difference {1,2,3} \ {2,3,4} is {1} , while, conversely, the set difference {2,3,4} \ {1,2,3} is {4} . When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams. 122 CHAPTER 34. SET THEORY

• Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (A ∪ B)\(A ∩ B) or (A \ B) ∪ (B \ A). • Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B. The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}. • Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .

Some basic sets of central importance are the empty set (the unique set containing no elements), the set of natural numbers, and the set of real numbers.

34.3 Some ontology

Main article: von Neumann universe A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set {{}} containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number α, known as its rank. The rank of a pure set X is defined to be the least upper bound of all successors of ranks of members of X. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α, the set Vα is defined to consist of all pure sets with rank less than α. The entire von Neumann universe is denoted V.

34.4 Axiomatic set theory

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell’s paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.[5] The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:

• Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), which includes the axiom of choice. Fragments of ZFC include: • Zermelo set theory, which replaces the axiom schema of replacement with that of separation; • , a small fragment of Zermelo set theory sufficient for the and finite sets; • Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement. • Sets and proper classes. These include Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse-Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC.

The above systems can be modified to allow urelements, objects that can be members of sets but that are not them- selves sets and do not have any members. The systems of New Foundations NFU (allowing urelements) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a “set of everything,” relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. 34.5. APPLICATIONS 123

Vω*ω ...

V3ω

V2ω ...

Vω+2 Vω+1

Vω ...

V5 V4 V3 V2 V1

V0

An initial segment of the von Neumann hierarchy.

Systems of , such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject. An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.

34.5 Applications

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (ax- iomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory. Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For 124 CHAPTER 34. SET THEORY

example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set. Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is like- wise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes human- written, computer‐verified derivations of more than 12,000 theorems starting from ZFC set theory, first order logic and propositional logic.

34.6 Areas of study

Set theory is a major area of research in mathematics, with many interrelated subfields.

34.6.1 Combinatorial set theory

Main article: Infinitary combinatorics

Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey’s theorem such as the Erdős–Rado theorem.

34.6.2 Descriptive set theory

Main article: Descriptive set theory

Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals. The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending (“relativizing”) it to make it more broadly applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.

34.6.3 Fuzzy set theory

Main article: Fuzzy set theory

In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of “tall people” is more flexible than a simple yes or no answer and can be a real number such as 0.75.

34.6.4 Inner model theory

Main article: Inner model theory

An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that 34.6. AREAS OF STUDY 125 the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent. The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).[6]

34.6.5 Large cardinals

Main article: Large cardinal property

A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory.

34.6.6 Determinacy

Main article: Determinacy

Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure.

34.6.7 Forcing

Main article: Forcing (mathematics)

Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. “forced”) by the construction and the original model. For example, Cohen’s construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.

34.6.8 Cardinal invariants

A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory. 126 CHAPTER 34. SET THEORY

34.6.9 Set-theoretic topology

Main article: Set-theoretic topology

Set-theoretic topology studies questions of that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

34.7 Objections to set theory as a foundation for mathematics

From set theory’s inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory’s earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. Ludwig Wittgenstein condemned set theory. He wrote that “set theory is wrong”, since it builds on the “nonsense” of fictitious symbolism, has “pernicious idioms”, and that it is nonsensical to talk about “all numbers”.[7] Wittgenstein’s views about the foundations of mathematics were later criticised by Georg Kreisel and , and investigated by Crispin Wright, among others. Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and theory.[8] Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces.[9] An active area of research is the univalent foundations arising from homotopy type theory. Here, sets may be defined as certain kinds of types, with universal properties of sets arising from higher inductive types. Principles such as the axiom of choice and the law of the excluded middle appear in a spectrum of different forms, some of which can be proven, others which correspond to the classical notions; this allows for a detailed discussion of the effect of these axioms on mathematics.[10][11]

34.8 See also

• Glossary of set theory

• Category theory

• List of set theory topics

• Relational model – borrows from set theory

34.9 Notes

[1] Cantor, Georg (1874), “Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen”, J. Reine Angew. Math. 77: 258–262, doi:10.1515/crll.1874.77.258

[2] Johnson, Philip (1972), A History of Set Theory, Prindle, Weber & Schmidt, ISBN 0-87150-154-6

[3] Bolzano, Bernard (1975), Berg, Jan, ed., Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., Vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich From- mann Verlag, p. 152, ISBN 3-7728-0466-7

[4] William Henry Young & Grace Chisholm Young (1906) Theory of Sets of Points, link from Internet Archive 34.10. FURTHER READING 127

[5] In his 1925, John von Neumann observed that “set theory in its first, “naive” version, due to Cantor, led to contradictions. These are the well-known antinomies of the set of all sets that do not contain themselves (Russell), of the set of all transfinte ordinal numbers (Burali-Forti), and the set of all finitely definable real numbers (Richard).” He goes on to observe that two “tendencies” were attempting to “rehabilitate” set theory. Of the first effort, exemplified by Bertrand Russell, Julius König, Hermann Weyl and L. E. J. Brouwer, von Neumann called the “overall effect of their activity . . . devastating”. With regards to the axiomatic method employed by second group composed of Zermelo, Abraham Fraenkel and Arthur Moritz Schoenflies, von Neumann worried that “We see only that the known modes of leading to the antinomies fail, but who knows where there are not others?" and he set to the task, “in the spirit of the second group”, to “produce, by means of a finite number of purely formal operations . . . all the sets that we want to see formed” but not allow for the antinomies. (All quotes from von Neumann 1925 reprinted in van Heijenoort, Jean (1967, third printing 1976), “From Frege to Gödel: A Source Book in Mathematical Logic, 1979–1931”, Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 (pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann’s 1925. [6] Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York: Springer-Verlag, p. 642, ISBN 978-3-540-44085-7, Zbl 1007.03002 [7] Wittgenstein, Ludwig (1975). Philosophical Remarks, §129, §174. Oxford: Basil Blackwell. ISBN 0631191305. [8] Ferro, A.; Omodeo, E. G.; Schwartz, J. T. (1980), “Decision procedures for elementary sublanguages of set theory. I. Multi-level syllogistic and some extensions”, Comm. Pure Appl. Math. 33 (5): 599–608, doi:10.1002/cpa.3160330503 [9] Saunders Mac Lane and Ieke Moerdijk (1992) Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Springer Verlag. [10] homotopy type theory in nLab [11] Homotopy Type Theory: Univalent Foundations of Mathematics. The Univalent Foundations Program. Institute for Ad- vanced Study.

34.10 Further reading

• Devlin, Keith, 1993. The Joy of Sets (2nd ed.). Springer Verlag, ISBN 0-387-94094-4 • Ferreirós, Jose, 2007 (1999). Labyrinth of Thought: A history of set theory and its role in modern mathematics. Basel, Birkhäuser. ISBN 978-3-7643-8349-7 • Johnson, Philip, 1972. A History of Set Theory. Prindle, Weber & Schmidt ISBN 0-87150-154-6 • Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. North-Holland, ISBN 0-444- 85401-0. • Potter, Michael, 2004. Set Theory and Its Philosophy: A Critical Introduction. Oxford University Press. • Tiles, Mary, 2004 (1989). The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise. Dover Publications. ISBN 978-0-486-43520-6

34.11 External links

• Foreman, Matthew, Akihiro Kanamori, eds. Handbook of Set Theory. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993). • Hazewinkel, Michiel, ed. (2001), “Axiomatic set theory”, Encyclopedia of Mathematics, Springer, ISBN 978- 1-55608-010-4 • Hazewinkel, Michiel, ed. (2001), “Set theory”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4 • Jech, Thomas (2002). "Set Theory", Stanford Encyclopedia of Philosophy. • Schoenflies, Arthur (1898). Mengenlehre in Klein’s encyclopedia. • Online books, and library resources in your library and in other libraries about set theory Chapter 35

Stanislaw Ulam

Stanisław Marcin Ulam (pronounced ['staɲiswaf 'martɕin͡ 'ulam]; 13 April 1909 – 13 May 1984) was a Polish- American mathematician. He participated in America’s Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures. Born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his D.Sc. in 1933 under the supervision of Kazimierz Kuratowski. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, where he worked to establish important results regarding ergodic theory. On 20 August 1939, he sailed for America for the last time with his 17-year-old brother Adam Ulam. He became an assistant professor at the University of Wisconsin–Madison in 1940, and a United States citizen in 1941. In October 1943, he received an invitation from Hans Bethe to join the Manhattan Project at the secret Los Alamos Laboratory in New Mexico. There, he worked on the hydrodynamic calculations to predict the behavior of the explosive lenses that were needed by an implosion-type weapon. He was assigned to Edward Teller's group, where he worked on Teller’s “Super” bomb for Teller and Enrico Fermi. After the war he left to become an associate professor at the University of Southern California, but returned to Los Alamos in 1946 to work on thermonuclear weapons. With the aid of a cadre of female "computers", including his wife Françoise Aron Ulam, he found that Teller’s “Super” design was unworkable. In January 1951, Ulam and Teller came up with the Teller–Ulam design, which is the basis for all thermonuclear weapons. Ulam considered the problem of nuclear propulsion of rockets, which was pursued by Project Rover, and proposed, as an alternative to Rover’s nuclear thermal rocket, to harness small nuclear explosions for propulsion, which became Project Orion. With Fermi and John Pasta, Ulam studied the Fermi–Pasta–Ulam problem, which became the in- spiration for the field of non-linear science. He is probably best known for realising that electronic computers made it practical to apply statistical methods to functions without known solutions, and as computers have developed, the Monte Carlo method has become a common and standard approach to many problems.

35.1 Poland

Ulam was born in Lemberg, Galicia, on 13 April 1909. At this time, Galicia was in the Kingdom of Galicia and Lodomeria of the Austro-Hungarian Empire, known to Poles as the Austrian partition. In 1918, it became part of the newly restored Poland, the Second Polish Republic, and the city took its Polish name again, Lwów.[1] The Ulams were a wealthy Polish Jewish family of bankers, industrialists, and other professionals. Ulam’s immediate family was “well-to-do but hardly rich”.[2] His father, Józef Ulam, was born in Lwów and was a lawyer,[1] and his mother, Anna (née Auerbach), was born in Stryj.[3] His uncle, Michał Ulam, was an architect, building contractor, and lumber industrialist.[4] From 1916 until 1918, Józef’s family lived temporarily in Vienna.[5] After they returned, Lwów became the epicenter of the Polish–Ukrainian War, during which the city experienced a Ukrainian siege.[1] In 1919, Ulam entered Lwów Gymnasium Nr. VII, from which he graduated in 1927.[6] He then studied mathematics at the Lwów Polytechnic Institute. Under the supervision of Kazimierz Kuratowski, he received his Master of Arts degree in 1932, and became a Doctor of Science in 1933.[5][7] At the age of 20, in 1929, he published his first

128 35.2. COMING TO AMERICA 129

The Scottish Café's building now houses the Universal Bank in Lviv, the present name of Lwów. paper Concerning Function of Sets in the journal Fundamenta Mathematicae.[7] From 1931 until 1935, he traveled to and studied in Wilno (Vilnius), Vienna, Zurich, Paris, and Cambridge, England, where he met G. H. Hardy and Subrahmanyan Chandrasekhar.[8] Along with Stanisław Mazur, Mark Kac, Włodzimierz Stożek, Kuratowski, and others, Ulam was a member of the Lwów School of Mathematics. Its founders were Hugo Steinhaus and Stefan Banach, who were professors at the University of Lwów. Mathematicians of this “school” met for long hours at the Scottish Café, where the problems they discussed were collected in the Scottish Book, a thick notebook provided by Banach’s wife. Ulam was a major contributor to the book. Of the 193 problems recorded between 1935 and 1941, he contributed 40 problems as a single author, another 11 with Banach and Mazur, and an additional 15 with others. In 1957, he received from Steinhaus a copy of the book, which had survived the war, and translated it into English.[9] In 1981, Ulam’s friend R. Daniel Maudlin published an expanded and annotated version.[10]

35.2 Coming to America

In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. In December of that year, Ulam sailed to America. At Princeton, he went to lectures and seminars, where he heard Oswald Veblen, James Alexander, and Albert Einstein. During a tea party at von Neumann’s house, he encountered G. D. Birkhoff, who suggested that he apply for a position with the Harvard Society of Fellows.[5] Following up on Birkhoff’s suggestion, Ulam spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts from 1936 to 1939, where he worked with John C. Oxtoby to establish results regarding ergodic theory. These appeared in Annals of Mathematics in 1941.[6][11] On 20 August 1939, in Gdynia, Józef Ulam, along with his brother Szymon, put his two sons, Stanislaw and 17 year old Adam, on a ship headed for America.[5] Two weeks later, the Germans invaded Poland. Within two months, the Germans completed their occupation of western Poland, and the Soviets invaded and occupied eastern Poland. Within two years, Józef Ulam and the rest of his family were victims of the Holocaust, Hugo Steinhaus was in hiding, 130 CHAPTER 35. STANISLAW ULAM

Kazimierz Kuratowski was lecturing at the underground university in Warsaw, Włodzimierz Stożek and his two sons had been killed in the massacre of Lwów professors, and the last problem had been recorded in the Scottish Book. Stefan Banach survived the Nazi occupation by feeding lice at Rudolf Weigl’s typhus research institute. In 1963, Adam Ulam, who had become an eminent kremlinologist at Harvard,[12] received a letter from George Volsky,[13] who hid in Józef Ulam’s house after deserting from the Polish army. This reminiscence gave a chilling account of Lwów’s chaotic scenes in late 1939.[14] In later life Ulam described himself as “an agnostic. Sometimes I muse deeply on the forces that are for me invisible. When I am almost close to the idea of God, I feel immediately estranged by the horrors of this world, which he seems to tolerate”.[15] In 1940, after being recommended by Birkhoff, Ulam became an assistant professor at the University of Wisconsin– Madison. Here, he became an United States citizen in 1941.[5] That year, he married Françoise Aron.[6] She had been a French exchange student at Mount Holyoke College, whom he met in Cambridge. They had one daughter, Claire. In Madison, Ulam met his friend and colleague C. J. Everett, with whom he would collaborate on a number of papers.[16]

35.3 Manhattan Project

Ulam’s ID badge photo from Los Alamos 35.4. POST WAR LOS ALAMOS 131

In early 1943, Ulam asked von Neumann to find him a war job. In October, he received an invitation to join an uniden- tified project near Santa Fe, New Mexico.[5] The letter was signed by Hans Bethe, who had been appointed as leader of the theoretical division of Los Alamos National Laboratory by Robert Oppenheimer, its scientific director.[17] Know- ing nothing of the area, he borrowed a New Mexico guide book. On the checkout card, he found the names of his Wis- consin colleagues, Joan Hinton, David Frisch, and Joseph McKibben, all of whom had mysteriously disappeared.[5] This was Ulam’s introduction to the Manhattan Project, which was America’s wartime effort to create the atomic bomb.[18]

35.3.1 Hydrodynamical calculations of implosion

A few weeks after Ulam reached Los Alamos in February 1944, the project experienced a crisis. In April, Emilio Segrè discovered that plutonium made in reactors would not work in a gun-type plutonium weapon like the "Thin Man", which was being developed in parallel with a uranium weapon, the "Little Boy" that was dropped on Hiroshima. This problem threatened to waste an enormous investment in new reactors at the Hanford site and to make slow separation of uranium isotopes the only way to prepare fissile material suitable for use in bombs. To respond, Op- penheimer implemented, in August, a sweeping reorganization of the laboratory to focus on development of an implosion-type weapon and appointed George Kistiakowsky head of the implosion department. He was a professor at Harvard and an expert on precise use of explosives.[19] The basic concept of implosion is to use chemical explosives to crush a chunk of fissile material into a critical mass, where neutron multiplication leads to a nuclear chain reaction, releasing a large amount of energy. Cylindrical im- plosive configurations had been studied by Seth Neddermeyer, but von Neumann, who had experience with shaped charges used in armor piercing ammunition, was a vocal advocate of spherical implosion driven by explosive lenses. He realized that the symmetry and speed with which implosion compressed the plutonium were critical issues,[19] and enlisted Ulam to help design lens configurations that would provide nearly spherical implosion. Within an implo- sion, because of enormous pressures and high temperatures, solid materials behave much like fluids. This meant that hydrodynamical calculations were needed to predict and minimize asymmetries that would spoil a nuclear detonation. Of these calculations, Ulam said:

The hydrodynamical problem was simply stated, but very difficult to calculate – not only in detail, but even in order of magnitude. In this discussion, I stressed pure pragmatism and the necessity to get a heuristic survey of the problem by simple-minded brute force, rather than by massive numerical work.[5]

Nevertheless, with the primitive facilities available at the time, Ulam and von Neumann did carry out numerical computations that led to a satisfactory design. This motivated their advocacy of a powerful computational capability at Los Alamos, which began during the war years,[20] continued through the cold war, and still exists.[21] Otto Frisch remembered Ulam as “a brilliant Polish topologist with a charming French wife. At once he told me that he was a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points!"[22]

35.3.2 Statistics of branching and multiplicative processes

Even the inherent statistical fluctuations of neutron multiplication within a chain reaction have implications with regard to implosion speed and symmetry. In November 1944, David Hawkins[23] and Ulam addressed this problem in a report entitled “Theory of Multiplicative Processes”.[24] This report, which invokes probability-generating functions, is also an early entry in the extensive literature on statistics of branching and multiplicative processes. In 1948, its scope was extended by Ulam and Everett.[25] Early in the Manhattan project, Enrico Fermi's attention was focused on the use of reactors to produce plutonium. In September 1944, he arrived at Los Alamos, shortly after breathing life into the first Hanford reactor, which had been poisoned by a xenon isotope.[26] Soon after Fermi’s arrival, Teller’s “Super” bomb group, of which Ulam was a part, was transferred to a new division headed by Fermi.[27] Fermi and Ulam formed a relationship that became very fruitful after the war.[28]

35.4 Post war Los Alamos

In September 1945, Ulam left Los Alamos to become an associate professor at the University of Southern California in Los Angeles. In January 1946, he suffered an acute attack of encephalitis, which put his life in danger, but which was 132 CHAPTER 35. STANISLAW ULAM

alleviated by emergency brain surgery. During his recuperation, many friends visited, including Nicholas Metropolis from Los Alamos and the famous mathematician Paul Erdős,[29] who remarked: “Stan, you are just like before.”[5] This was encouraging, because Ulam was concerned about the state of his mental faculties, for he had lost the ability to speak during the crisis. Another friend, Gian-Carlo Rota, asserted in a 1987 article that the attack changed Ulam’s personality; afterwards, he turned from rigorous pure mathematics to more speculative conjectures concerning the application of mathematics to physics and biology.[30] This assertion was not accepted by Françoise Aron Ulam.[31] By late April 1946, Ulam had recovered enough to attend a secret conference at Los Alamos to discuss thermonuclear weapons. Those in attendance included Ulam, von Neumann, Metropolis, Teller, Stan Frankel, and others. Through- out his participation in the Manhattan Project, Teller’s efforts had been directed toward the development of a “super” weapon based on nuclear fusion, rather than toward development of a practical fission bomb. After extensive dis- cussion, the participants reached a consensus that his ideas were worthy of further exploration. A few weeks later, Ulam received an offer of a position at Los Alamos from Metropolis and Robert D. Richtmyer, the new head of its theoretical division, at a higher salary, and the Ulams returned to Los Alamos.[32]

35.4.1 Monte Carlo method

Late in the war, under the sponsorship of von Neumann, Frankel and Metropolis began to carry out calculations on the first general-purpose electronic computer, the ENIAC. Shortly after returning to Los Alamos, Ulam participated in a review of results from these calculations.[33] Earlier, while playing solitaire during his recovery from surgery, Ulam had thought about playing hundreds of games to estimate statistically the probability of a successful outcome.[34] With ENIAC in mind, he realized that the availability of computers made such statistical methods very practical. John von Neumann immediately saw the significance of this insight. In March 1947 he proposed a statistical approach to the problem of neutron diffusion in fissionable material.[35] Because Ulam had often mentioned his uncle, Michał Ulam, “who just had to go to Monte Carlo” to gamble, Metropolis dubbed the statistical approach “The Monte Carlo method".[33] Metropolis and Ulam published the first unclassified paper on the Monte Carlo method in 1949.[36] Fermi, learning of Ulam’s breakthrough, devised an analog computer known as the Monte Carlo trolley, later dubbed the FERMIAC. The device performed a mechanical simulation of random diffusion of neutrons. As computers improved in speed and programmability, these methods became more useful. In particular, many Monte Carlo calculations carried out on modern massively parallel supercomputers are embarrassingly parallel applications, whose results can be very accurate.[21]

35.4.2 Teller–Ulam design

On 29 August 1949, the Soviet Union tested its first fission bomb, the RDS-1. Created under the supervision of Lavrentiy Beria, who sought to duplicate the American effort, this weapon was nearly identical to Fat Man, for its design was based on information provided by spies Klaus Fuchs, Theodore Hall, and David Greenglass. In response, on 31 January 1950, President Harry S. Truman announced a crash program to develop a fusion bomb.[37] To advocate an aggressive development program, Ernest Lawrence and Luis Alvarez came to Los Alamos, where they conferred with Norris Bradbury, the laboratory director, and with George Gamow, Edward Teller, and Ulam. Soon, these three became members of a short-lived committee appointed by Bradbury to study the problem, with Teller as chairman.[5] At this time, research on the use of a fission weapon to create a fusion reaction had been ongoing since 1942, but the design was still essentially the one originally proposed by Teller. His concept was to put tritium and/or deuterium in close proximity to a fission bomb, with the hope that the heat and intense flux of neutrons released when the bomb exploded, would ignite a self-sustaining fusion reaction. Reactions of these isotopes of hydrogen are of interest because the energy per unit mass of fuel released by their fusion is much larger than that from fission of heavy nuclei.[38] Because the results of calculations based on Teller’s concept were discouraging, many scientists believed it could not lead to a successful weapon, while others had moral and economic grounds for not proceeding. Consequently, several senior people of the Manhattan Project opposed development, including Bethe and Oppenheimer.[39] To clarify the situation, Ulam and von Neumann resolved to do new calculations to determine whether Teller’s approach was feasible. To carry out these studies, von Neumann decided to use electronic computers: ENIAC at Aberdeen, a new computer, MANIAC, at Princeton, and its twin, which was under construction at Los Alamos. Ulam enlisted Everett to follow a completely different approach, one guided by physical intuition. Françoise Ulam was one of a cadre of women "computers" who carried out laborious and extensive computations of thermonuclear scenarios on mechanical calculators, supplemented and confirmed by Everett’s slide rule. Ulam and Fermi collaborated on further 35.4. POST WAR LOS ALAMOS 133

Stan Ulam Holding the FERMIAC

analysis of these scenarios. The results showed that, in workable configurations, a thermonuclear reaction would not ignite, and if ignited, it would not be self-sustaining. Ulam had used his expertise in Combinatorics to analyze the chain reaction in deuterium, which was much more complicated than the ones in uranium and plutonium, and he concluded that no self-sustaining chain reaction would take place at the (low) densities that Teller was considering.[40] In late 1950, these conclusions were confirmed by von Neumann’s results.[31][41] In January 1951, Ulam had another idea: to channel the mechanical shock of a nuclear explosion so as to compress the fusion fuel. On the recommendation of his wife,[31] Ulam discussed this idea with Bradbury and Mark before he told Teller about it.[42] Almost immediately, Teller saw its merit, but noted that soft X-rays from the fission bomb would compress the thermonuclear fuel more strongly than mechanical shock and suggested ways to enhance this effect. On 9 March 1951, Teller and Ulam submitted a joint report describing these innovations.[43] A few weeks later, Teller suggested placing a fissile rod or cylinder at the center of the fusion fuel. The detonation of this “spark plug”[44] would help to initiate and enhance the fusion reaction. The design based on these ideas, called staged radiation implosion, has become the standard way to build thermonuclear weapons. It is often described as the "Teller–Ulam design".[45] 134 CHAPTER 35. STANISLAW ULAM

Ivy Mike, the first full test of the Teller–Ulam design (a staged fusion bomb), with a yield of 10.4 megatons on 1 November 1952

In September 1951, after a series of differences with Bradbury and other scientists, Teller resigned from Los Alamos, and returned to the University of Chicago.[46] At about the same time, Ulam went on leave as a visiting professor at Harvard for a semester.[47] Although Teller and Ulam submitted a joint report on their design[43] and jointly applied for a patent on it,[18] they soon became involved in a dispute over who deserved credit.[42] After the war, Bethe returned to Cornell University, but he was deeply involved in the development of thermonuclear weapons as a consultant. In 1954, he wrote an article on the history of the H-bomb,[48] which presents his opinion that both men contributed very significantly to the breakthrough. This balanced view is shared by others who were involved, including Mark and Fermi, but Teller persistently attempted to downplay Ulam’s role.[49] “After the H-bomb was made,” Bethe recalled, “reporters started to call Teller the father of the H-bomb. For the sake of history, I think it is more precise to say that Ulam is the father, because he provided the seed, and Teller is the mother, because he remained with the child. As for me, I guess I am the midwife.”[50] With the basic fusion reactions confirmed, and with a feasible design in hand, there was nothing to prevent Los Alamos from testing a thermonuclear device. On 1 November 1952, the first thermonuclear explosion occurred when Ivy Mike was detonated on Enewetak Atoll, within the US Pacific Proving Grounds. This device, which used liquid deuterium as its fusion fuel, was immense and utterly unusable as a weapon. Nevertheless, its success validated the Teller–Ulam design, and stimulated intensive development of practical weapons.[47]

35.4.3 Fermi–Pasta–Ulam problem

Main article: Fermi–Pasta–Tsingou-Ulam problem

When Ulam returned to Los Alamos, his attention turned away from weapon design and toward the use of computers to investigate problems in physics and mathematics. With John Pasta, who helped Metropolis to bring MANIAC on line in March 1952, he explored these ideas in a report “Heuristic Studies in Problems of Mathematical Physics on High Speed Computing Machines”, which was submitted on 9 June 1953. It treated several problems that cannot be addressed within the framework of traditional analytic methods: billowing of fluids, rotational motion in gravitating 35.4. POST WAR LOS ALAMOS 135

The Sausage device of Mike nuclear test (yield 10.4 Mt) on Enewetak Atoll. The test was part of the Operation Ivy. The Sausage was the first true H-Bomb ever tested, meaning the first thermonuclear device built upon the Teller-Ulam principles of staged radiation implosion. systems, magnetic lines of force, and hydrodynamic instabilities.[51] Soon, Pasta and Ulam became experienced with electronic computation on MANIAC, and by this time, Enrico Fermi had settled into a routine of spending academic years at the University of Chicago and summers at Los Alamos. During these summer visits, Pasta and Ulam joined him to study a variation of the classic problem of a string of masses held together by springs that exert forces linearly proportional to their displacement from equilibrium. Fermi proposed to add to this force a nonlinear component, which could be chosen to be proportional to either the square or cube of the displacement, or to a more complicated “broken linear” function. This addition is the key element of the Fermi–Pasta–Ulam problem, which is often designated by the abbreviation FPU.[52][53] A classical spring system can be described in terms of vibrational modes, which are analogous to the harmonics that occur on a stretched violin string. If the system starts in a particular mode, vibrations in other modes do not develop. With the nonlinear component, Fermi expected energy in one mode to transfer gradually to other modes, and even- tually, to be distributed equally among all modes. This is roughly what began to happen shortly after the system was initialized with all its energy in the lowest mode, but much later, essentially all the energy periodically reappeared in the lowest mode.[53] This behavior is very different from the expected equipartition of energy. It remained mysterious until 1965, when Kruskal and Zabusky showed that, after appropriate mathematical transformations, the system can be described by the Korteweg–de Vries equation, which is the prototype of nonlinear partial differential equations that have soliton solutions. This means that FPU behavior can be understood in terms of solitons.[54]

35.4.4 Nuclear propulsion

Starting in 1955, Ulam and Frederick Reines considered nuclear propulsion of aircraft and rockets.[55] This is an attractive possibility, because the nuclear energy per unit mass of fuel is a million times greater than that available from chemicals. From 1955 to 1972, their ideas were pursued during Project Rover, which explored the use of nuclear reactors to power rockets.[56] In response to a question by Senator John O. Pastore at a congressional committee hearing on “Outer Space Propulsion by Nuclear Energy”, on January 22, 1958, Ulam replied that “the future as a 136 CHAPTER 35. STANISLAW ULAM

An artist’s conception of the NASA reference design for the Project Orion spacecraft powered by nuclear propulsion

whole of mankind is to some extent involved inexorably now with going outside the globe.”[57] Ulam and C. J. Everett also proposed, in contrast to Rover’s continuous heating of rocket exhaust, to harness small nuclear explosions for propulsion.[58] Project Orion was a study of this idea. It began in 1958 and ended in 1965, after the Partial Nuclear Test Ban Treaty of 1963 banned nuclear weapons tests in the atmosphere and in space.[59] Work on this project was spearheaded by physicist Freeman Dyson, who commented on the decision to end Orion in his article, “Death of a Project”.[60] Bradbury appointed Ulam and John H. Manley as research advisors to the laboratory director in 1957. These newly created positions were on the same administrative level as division leaders, and Ulam held his until he retired from Los Alamos. In this capacity, he was able to influence and guide programs in many divisions: theoretical, physics, chemistry, metallurgy, weapons, health, Rover, and others.[56] In addition to these activities, Ulam continued to publish technical reports and research papers. One of these intro- duced the Fermi–Ulam model, an extension of Fermi’s theory of the acceleration of cosmic rays.[61] Another, with Paul Stein and Mary Tsingou, titled “Quadratic Transformations”, was an early investigation of chaos theory and is considered the first published use of the phrase "chaotic behavior".[62][63]

35.5 Return to academia

During his years at Los Alamos, Ulam was a visiting professor at Harvard from 1951 to 1952, MIT from 1956 to 1957, the University of California, San Diego, in 1963, and the University of Colorado at Boulder from 1961 to 1962 and 1965 to 1967. In 1967, the last of these positions became permanent, when Ulam was appointed as professor and Chairman of the Department of Mathematics at Boulder, Colorado. He kept a residence in Santa Fe, New Mexico, which made it convenient to spend summers at Los Alamos as a consultant.[64] 35.5. RETURN TO ACADEMIA 137

When the positive integers are arrayed along the Ulam spiral, prime numbers, represented by dots, tend to collect along diagonal lines.

In Colorado, where he rejoined his friends Gamow, Richtmyer, and Hawkins, Ulam’s research interests turned toward biology. In 1968, recognizing this emphasis, the University of Colorado School of Medicine appointed Ulam as Professor of Biomathematics, and he held this position until his death. With his Los Alamos colleague Robert Schrandt he published a report, “Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutionary Processes”, which applied his earlier ideas on branching processes to biological inheritance.[65] Another, report, with William Beyer, Temple F. Smith, and M. L. Stein, titled “Metrics in Biology”, introduced new ideas about biometric distances.[66] When he retired from Colorado in 1975, Ulam had begun to spend winter semesters at the University of Florida, where he was a graduate research professor. Except for sabbaticals at the University of California, Davis from 1982 to 1983, and at Rockefeller University from 1980 to 1984,[64] this pattern of spending summers in Colorado and Los Alamos and winters in Florida continued until Ulam died of an apparent heart attack in Santa Fe on 13 May 1984.[67] Paul Erdős noted that “he died suddenly of heart failure, without fear or pain, while he could still prove and conjecture.”[29] In 1987, Françoise Ulam deposited his papers with the American Philosophical Society Library in Philadelphia.[68] She continued to live in Santa Fe until she died on 30 April 2011, at the age of 93. Both Françoise and her husband are buried with her French family in Montmartre Cemetery in Paris.[69] 138 CHAPTER 35. STANISLAW ULAM

35.6 Impact and legacy

From the publication of his first paper as a student in 1929 until his death, Ulam was constantly writing on mathe- matics. The list of Ulam’s publications includes more than 150 papers.[6] Topics represented by a significant number of papers are: set theory (including measurable cardinals and abstract measures), topology, transformation theory, ergodic theory, group theory, projective algebra, number theory, combinatorics, and graph theory.[70] In March 2009, the Mathematical Reviews database contained 697 papers with the name “Ulam”.[71] Notable results of this work are: With his pivotal role in the development of thermonuclear weapons, Stanislaw Ulam changed the world. According to Françoise Ulam: “Stan would reassure me that, barring accidents, the H-bomb rendered nuclear war impossible.”[31] In 1980, Ulam and his wife appeared in the television documentary The Day After Trinity.[72] The Monte Carlo method has become a ubiquitous and standard approach to computation, and the method has been applied to a vast number of scientific problems.[73] In addition to problems in physics and mathematics, the method has been applied to finance, social science,[74] environmental risk assessment,[75] linguistics,[76] radiation therapy,[77] and sports.[78] The Fermi–Pasta–Ulam problem is credited not only as “the birth of experimental mathematics”,[53] but also as inspiration for the vast field of Nonlinear Science. In his Lilienfeld Prize lecture, David K. Campbell noted this relationship and described how FPU gave rise to ideas in chaos, solitons, and dynamical systems.[79] In 1980, Donald Kerr, laboratory director at Los Alamos, with the strong support of Ulam and Mark Kac,[80] founded the Center for Nonlinear Studies (CNLS).[81] In 1985, CNLS initiated the Stanislaw M. Ulam Distinguished Scholar program, which provides an annual award that enables a noted scientist to spend a year carrying out research at Los Alamos.[82] The fiftieth anniversary of the original FPU paper was the subject of the March 2005 issue of the journal Chaos,[83] and the topic of the 25th Annual International Conference of CNLS.[84] The University of Southern Mississippi and the University of Florida supported the Ulam Quarterly,[85] which was active from 1992 to 1996, and which was one of the first online mathematical journals.[86] Florida’s Department of Mathematics has sponsored, since 1998, the annual Ulam Colloquium Lecture,[87] and in March 2009, the Ulam Centennial Conference.[88] Ulam’s work on non-Euclidean distance metrics in the context of molecular biology made a significant contribution to sequence analysis[89] and his contributions in theoretical biology are considered watersheds in the development of cellular automata theory, population biology, pattern recognition, and biometrics generally. Colleagues noted that some of his greatest contributions were in clearly identifying problems to be solved and general techniques for solving them.[90] In 1987, Los Alamos issued a special issue of its Science publication, which summarized his accomplishments,[91] and which appeared, in 1989, as the book From Cardinals to Chaos. Similarly, in 1990, the University of California Press issued a compilation of mathematical reports by Ulam and his Los Alamos collaborators: Analogies Between Analogies.[92] During his career, Ulam was awarded honorary degrees by the Universities of New Mexico, Wisconsin, and Pittsburgh.[5]

35.7 Bibliography

• Kac, Mark; Ulam, Stanisław (1968). Mathematics and Logic: Retrospect and Prospects. New York: Praeger. ISBN 978-0-486-67085-0. OCLC 24847821.

• Ulam, Stanisław (1974). Beyer, W. A.; Mycielski and, J.; Rota, G.-C., eds. Sets, Numbers, and Universes: selected works. Mathematicians of Our Time 9. The MIT Press, Cambridge, Mass.-London. ISBN 978-0- 262-02108-1. MR 0441664.

• Ulam, Stanisław (1960). A Collection of Mathematical Problems’. New York: Interscience Publishers. OCLC 526673.

• Ulam, Stanisław (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. ISBN 978-0- 684-14391-0. OCLC 1528346. (autobiography).

• Ulam, Stanisław (1986). Science, Computers, and People: From the Tree of Mathematics. Boston: Birkhauser. ISBN 978-3-7643-3276-1. OCLC 11260216. 35.8. SEE ALSO 139

An animation demonstrating the lucky number sieve. The numbers in red are lucky numbers

• Ulam, Stanisław; Ulam, Françoise (1990). Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and his Los Alamos Collaborators. Berkeley: University of California Press. ISBN 978-0-520-05290-1. OCLC 20318499.

35.8 See also

• Kuratowski–Ulam theorem 140 CHAPTER 35. STANISLAW ULAM

35.9 References

[1] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 9–15. ISBN 9780684143910. OCLC 1528346.

[2] Ulam, Adam Bruno (2002). Understanding the Cold War: a historian’s personal reflections. New Brunswick, NJ: Transac- tion Publishers. p. 19. ISBN 9780765808851. OCLC 48122759. Retrieved 28 December 2011.

[3] Ulam, Molly (June 25, 2000). “Ulam Family of Lwow; Auerbachs of Vienna”. Genforum. Retrieved 10 October 2011.

[4] “Genealogy of Michael Ulam”. GENi. 24 May 2011. Retrieved 12 October 2011.

[5] Ulam, Francoise (1987). “Vita” (PDF). Excerpts from Adventures of a Mathematician”. Los Alamos National Laboratory. Retrieved 7 October 2011.

[6] Ciesielski, Kryzystof; Thermistocles Rassias (2009). “On Stan Ulam and His Mathematics” (PDF). Australian Journal of Mathematical Analysis and Applications. Retrieved 10 October 2011. v 6, nr 1, pp 1-9, 2009

[7] Andrzej M. Kobos (1999). “Mędrzec większy niż życie” [A Sage Greater Than Life]. Zwoje (in Polish) 3 (16). Retrieved 10 May 2013.

[8] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 56–60. ISBN 9780684143910. OCLC 1528346.

[9] Ulam, Stanislaw (November 2002). “Preface to the “Scottish Book"". Turnbull WWW Server. School of Mathematical and Computational Sciences University of St Andrews. Retrieved 11 September 2012.

[10] Maudlin, R. Daniel (1981). The Scottish Book. Birkhauser. p. 268. ISBN 9783764330453. OCLC 7553633. Retrieved 4 December 2011.

[11] “Obituary for John C, Oxtoby”. The New York Times. 5 January 1991. Retrieved 10 October 2011.

[12] “Obituary for Adam Ulam”. Harvard University Gazette. 6 April 2000. Retrieved 10 October 2011.

[13] Volsky, George (23 December 1963). “Letter about Jozef Ulam”. Anxiously from Lwow. Adam Ulam. Retrieved 24 May 2013.

[14] “Lwow lives on at Leopolis Press”. The Hook. 14 November 2002. Retrieved 10 October 2011.

[15] Budrewicz/, Olgierd (1977). The melting-pot revisited: twenty well-known Americans of Polish background. Interpress. p. 36. Retrieved 11 September 2012.

[16] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 125–130, 174. ISBN 9780684143910. OCLC 1528346.

[17] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 143–147. ISBN 9780684143910. OCLC 1528346.

[18] “Staff biography of Stanislaw Ulam”. Los Alamos National Laboratory. Retrieved 22 October 2011.

[19] Hoddeson, Lillian; Henriksen, Paul W.; Meade, Roger A.; Westfall, Catherine L. (1993). Critical Assembly: A Technical History of Los Alamos During the Oppenheimer Years, 1943–1945. New York: Cambridge University Press. pp. 130–137. ISBN 0-521-44132-3. OCLC 26764320.

[20] “Supercomputing”. History @ Los Alamos. Los Alamos National Laboratory. Retrieved 24 October 2011.

[21] “From Calculators to Computers”. History @ Los Alamos. Los Alamos National Laboratory. Retrieved 24 October 2011.

[22] Frisch, Otto (April 1974). “Somebody Turned the Sun on with a Switch”. Bulletin of the Atomic Scientists 30 (4): 17. Retrieved May 29, 2013.

[23] Lehmann, Christopher (4 March 2002). “Obituary of David Hawkins”. The New York Times. Retrieved 14 October 2011.

[24] Hawkins, D.; S. Ulam (14 November 1944). “Theory of Multiplicative Processes” (PDF). LANL report LA-171. Retrieved 13 October 2011.

[25] Ulam, S.; C. J, Everett (7 June 1948). “Multiplicative Systems in Several Variables I, II, III”. LANL reports. University of California Press. Retrieved 13 October 2011.

[26] Hewlett, Richard G.; Anderson, Oscar E. (1962). The New World, 1939–1946 (PDF). University Park: Pennsylvania State University Press. pp. 304–307. ISBN 0-520-07186-7. OCLC 637004643. 35.9. REFERENCES 141

[27] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 152–153. ISBN 9780684143910. OCLC 1528346.

[28] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 162–157. ISBN 9780684143910. OCLC 1528346.

[29] Erdos, Paul (1985). “Ulam, the man and the mathematician” (PDF). J. Graph Theory, v 9, p445-449. Retrieved 10 October 2011.

[30] Rota, Gian-Carlo. “Stan Ulam: The Lost Cafe” (PDF). Los Alamos Science, No 15, 1987. Retrieved 22 October 2011.

[31] Ulam, Françoise (1991). Postscript to Adventures of a Mathematician. Berkeley, CA: University of California. ISBN 0-520-07154-9.

[32] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 184–187. ISBN 9780684143910. OCLC 1528346.

[33] Metropolis, Nicholas (1987). “The Beginnings of the Monte Carlo Method” (PDF). Los Alamos Science, No 15. Retrieved 22 October 2011.

[34] Eckhardt, Roger (1987). “Stan Ulam, John von Neumann, and the Monte Carlo method” (PDF). Los Alamos Science, No 15. Retrieved 22 October 2011.

[35] Richtmyer, D.; J. Pasta and S. Ulam (9 April 1947). “Statistical Methods in Neutron Diffusion” (PDF). LANL report LAMS-551. Retrieved 23 October 2011.

[36] Metropolis, Nicholas; Stanislaw Ulam (1949). “The Monte Carlo method” (PDF). Journal of the American Statistical Association 44: 335–341. doi:10.2307/2280232. PMID 18139350. Retrieved 21 November 2011.

[37] Hewlett, Richard G.; Duncan, Francis (1969). Atomic Shield, Volume II, 1947–1952. A History of the United States Atomic Energy Commission. University Park, Pennsylvania: Pennsylvania State University Press. pp. 406–409. ISBN 0-520-07187-5.

[38] Rhodes, Richard (1995). Dark Sun: The Making of the Hydrogen Bomb. New York: Simon & Schuster. p. 248. ISBN 0-684-80400-X.

[39] Hewlett, Richard G.; Duncan, Francis (1969). Atomic Shield, 1947–1952. A History of the United States Atomic Energy Commission. University Park: Pennsylvania State University Press. pp. 380–385. ISBN 0-520-07187-5. OCLC 3717478.

[40] Peter Galison (1996). “5: Computer Simulations and the Trading Zone”. In Peter Galison, David J. Stump. The Disunity of Science: Boundaries, Contexts, and Power. Stanford University Press. p. 135.

[41] Rhodes, Richard (1995). Dark Sun: The Making of the Hydrogen Bomb. New York: Simon & Schuster. pp. 422–424. ISBN 0-684-80400-X.

[42] “Staff biography of J. Carson Mark”. Los Alamos National Laboratory. Retrieved 22 October 2011.

[43] Teller, E.; S. Ulam (9 March 1951). “Heterocatalytic Detonations” (PDF). LANL report LAMS-1225. Retrieved 2 Novem- ber 2011.

[44] Teller, E. (4 April 1951), “A New Thermonuclear device”, Technical Report LAMS-1230, Los Alamos National Laboratory

[45] Rhodes, Richard (1995). Dark Sun: The Making of the Hydrogen Bomb. New York: Simon & Schuster. pp. 455–464. ISBN 0-684-80400-X.

[46] Hewlett, Richard G.; Duncan, Francis (1969). Atomic Shield, 1947–1952. A History of the United States Atomic Energy Commission. University Park: Pennsylvania State University Press. pp. 554–556. ISBN 0-520-07187-5. OCLC 3717478.

[47] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 220=224. ISBN 9780684143910. OCLC 1528346.

[48] Bethe, Hans A. (Fall 1982). “Reprinting of 1954 article: Comments on the History of the H-Bomb” (PDF). Los Alamos Science, No 6. Los Alamos National Laboratory. Retrieved 3 November 2011.

[49] Uchii, Soshichi (22 July 2003). “Review of Edward Teller’s Memoirs”. PHS Newsletter 52. Retrieved 13 August 2012.

[50] Schweber, S. S. (2000). In the Shadow of the Bomb: Bethe, Oppenheimer, and the Moral Responsibility of the Scientist. Princeton: Princeton University Press. p. 166. ISBN 978-0-691-04989-2.

[51] Pasta, John; S. Ulam (9 March 1953). “Heuristic studies in problems of mathematical physics” (PDF). LANL report LA- 1557. Retrieved 21 November 2011. 142 CHAPTER 35. STANISLAW ULAM

[52] Fermi, E.; J. Pasta and S. Ulam (May 1955). “Studies of Nonlinear Problems I” (PDF). LANL report LA-1940. Retrieved 21 November 2011.

[53] Porter, Mason A.; Norman J, Zabusky, Bambi Hu and David K. Campbell (May–Jun 2009). “Fermi, Pasta, Ulam and the Birth of Experimental Mathematics” (PDF). American Scientist 97 (3): 214–221. doi:10.1511/2009.78.214. Retrieved 20 November 2011.

[54] “Focus: Landmarks—Computer Simulations Led to Discovery of Solitons”. Physics 6 (15). February 8, 2013. doi:10.1103/Physics.6.15.

[55] Longmier, C.; F. Reines and S. Ulam (August 1955). “Some Schemes for Nuclear Propulsion” (PDF). LANL report LAMS- 2186. Retrieved 24 November 2011.

[56] Ulam, S. M (1983). Adventures of a Mathematician. New York: Charles Scribner’s Sons. pp. 249–250. ISBN 9780684143910. OCLC 1528346.

[57] Schreiber, R. E.; Ulam, Stanislaw M.; Bradbury, Norris (1958). “US Congress, Joint Committee on Atomic Energy: hearing on 22 January 1958”. Outer Space Propulsion by Nuclear Energy. US Government Printing Office. p. 47. Retrieved 25 November 2011.

[58] Everett, C. J.; S. M. Ulam (August 1955). “On a Method of Propulsion of Projectiles by Means of External Nuclear Explosions” (PDF). LANL report LAMS-1955. Retrieved 24 November 2011.

[59] “History of Project Orion”. The Story of Orion. OrionDrive.com. 2008–2009. Retrieved 7 October 2011.

[60] Dyson, Freeman (9 July 1965). “Death of a Project”. Science 149 (3680): 141–144. doi:10.1126/science.149.3680.141.

[61] Ulam, S. M. (1961), “On Some Statistical Properties of Dynamical Systems”, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley CA, v 3, p 315: University of California Press

[62] Abraham, Ralph (9 July 2011). “Image Entropy for Discrete Dynamical Systems” (PDF). University of California, Santa Cruz. Retrieved 30 May 2013.

[63] Stein, P. R.; Stanislaw M. Ulam (March 1959). “Quadratic Transformations. Part I” (PDF). LANL report LA-2305. Los Alamos National Laboratory. Retrieved 26 November 2011.

[64] “Stanislaw Ulam”. American Institute of Physics. Retrieved 14 May 2013.

[65] Schrandt, Robert G.; Stanislaw M. Ulam (December 1970). “Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutaionary Processes” (PDF). LANL report LA-4246. Los Alamos National Laboratory. Retrieved 26 November 2011.

[66] Beyer, William A.; Temple F. Smith; M. L. Stein; Stanislaw M. Ulam (August 1972). “Metrics in Biology, an Introduction” (PDF). LANL report LA-4973. Los Alamos National Laboratory. Retrieved 26 November 2011.

[67] Sullivan, Walter (15 May 1984). “Stanislaw Ulam, Theorist on Hydrogen Bomb”. New York Times. Retrieved 30 May 2013.

[68] “Stanislaw M. Ulam Papers”. American Philosophical Society. Retrieved 14 May 2013.

[69] “Françoise Ulam Obituary”. Santa Fe, New Mexican. 30 April 2011. Retrieved 12 December 2011.

[70] “Publications of Stanislaw M. Ulam” (PDF). Los Alamos Science, No 15, 1987. Los Alamos National Laboratory. Retrieved 6 December 2011.

[71] “Search for “Ulam” on AMS website”. American Mathematical Society. Retrieved 10 December 2011.

[72] The Day After Trinity at the Internet Movie Database

[73] Eckhardt, Roger (Special Issue, 1987). “Stan Ulam, John von Neumann, and the Monte Carlo Method” (PDF). Los Alamos Science. Los Alamos National Laboratory. Retrieved 23 May 2013. Check date values in: |date= (help)

[74] Casey, Thomas M. (June 2011). “Course description:Monte Carlo Methods for Social Scientists”. Inter-University Con- sortium for Political and Social Research. University of Michigan. Retrieved 9 December 2011.

[75] Poulter, Susan R. (Winter 1998). “Monte Carlo Simulation in Environmental Risk Assessment” (PDF). Risk:Health, Safety, & Environment. University of New Hampshire. Retrieved 13 September 2012.

[76] Klein, Sheldon (23 May 1966). “Historical Change in Language Using Monte CarloTechniques” (PDF). Mechanical Trans- lation and Computational Linguistics 9 (3 and 4): 67–81. Retrieved 9 December 2011. 35.10. EXTERNAL LINKS 143

[77] Earl, M. A.; L. M. Ma (12 March 2002). “Dose Enhancement of electron beams subject to external magnetic fields: A Monte Carlo Study”. Medical Physics 29: 484–492. doi:10.1118/1.1461374. Retrieved 9 December 2011.

[78] Ludwig, John (November 2011). “A Monte Carlo Simulation of the Big10 race”. ludwig.com. Retrieved 9 December 2011.

[79] Campbell, Donald H. (17 March 2010). “The Birth of Nonlinear Science” (PDF). Americal Physical Society. Retrieved 8 December 2011.

[80] “CNLS: apprecion of Martin Kruskal and Alwyn Scott”. Los Alamos National Laboratory. 2007. Retrieved 8 December 2011.

[81] “History of the Center for Nonlinear Studies”. Los Alamos National Laboratory. Retrieved 8 December 2011.

[82] “Ulam Scholars at CNLS”. Los Alamos National Laboratory. Retrieved 8 December 2011.

[83] “Focus-Issue: The Fermi-Pasta-Ulam Problem-The-First-50-Years”. Chaos 15 (1). March 2005. Retrieved 9 December 2011.

[84] “50 Years of the Fermi-Pasta-Ulam Problem: Legacy, Impact, and Beyond”. CLNS 25th International Conference. Los Alamos National Laboratory. May 16–20, 2005. Retrieved 9 December 2011.

[85] “Home Page for Ulam Quarterly”. University of Florida. Retrieved 24 December 2011.

[86] Dix, Julio G. (June 25–27, 2004), “Some Aspects of Running a Free Electronic Journal” (PDF), in Becker, Hans, New Developments in Electronic Publishing, Stockholm: European Congress of Mathematicians; ECM4 Satellite Conference, pp. 41–43, ISBN 978-3-88127-107-3, retrieved 5 January 2013

[87] “List of Ulam Colloquium Speakers”. University of Florida, Dept. of Mathematics. Retrieved 24 December 2011.

[88] “Ulam Centennial Conference”. University of Florida. March 10–11, 2009. Retrieved 24 December 2011.

[89] Goad, Walter B (1987). “Sequence Analysis: Contributions of Ulam to Molecular Genetics” (PDF). Los Alamos Science. Los Alamos National Laboratory. Retrieved 28 December 2011.

[90] Beyer, William A.; Peter H. Sellers; Michael S. Waterman (1985). “Stanislaw M. Ulam’s Contributions to Theoretical Biology” (PDF). Letters in Mathematical Physics 10: 231–242. doi:10.1007/bf00398163. Retrieved 5 December 2011.

[91] Cooper, Necia Grant. “Stanislaw Ulam 1909–1984”. Los Alamos Science, No 15, 1987. Los Alamos National Laboratory. Retrieved 6 December 2011.

[92] Ulam, S. M. (1990). A. R. Bednarek and Françoise Ulam, ed. Analogies Between Analogies. Berkeley: University of California Press. ISBN 0-520-05290-0. Retrieved 24 December 2011.

35.10 External links

• “Publications of Stanislaw M. Ulam” (PDF). Los Alamos Science. Special Issue 1987. ISSN 0273-7116. Check date values in: |date= (help) Chapter 36

Stefan Banach

Stefan Banach ([ˈstɛfan ˈbanax]; March 30, 1892 – August 31, 1945) was a Polish mathematician. He is generally considered to have been one of the 20th century’s most important and influential mathematicians. Banach was one of the founders of modern functional analysis and one of the original members of the Lwów School of Mathematics. His major work was the 1932 book, Théorie des opérations linéaires (Theory of Linear Operations), the first monograph on the general theory of functional analysis. Born in Kraków, Banach enrolled in the Henryk Sienkiewicz Gymnasium, a secondary school, and worked on math- ematics problems with his friend Witold Wiłkosz. After graduating in 1910, Banach and Wiłkosz moved to Lwów. However, Banach returned to Kraków during World War I, and during this time he met and befriended Hugo Stein- haus. After Banach solved mathematical problems which Steinhaus considered difficult, he and Steinhaus published their first joint work. Along with several other mathematicians, Banach formed a society for mathematicians in 1919. In 1920, after Poland had in 1918 regained independence, Banach was given an assistantship at Jagiellonian University. He soon became a professor at the Lwów Polytechnic and a member of the Polish Academy of Learning. Later Banach organized the “Lwów School of Mathematics”. Around 1929 he began writing Théorie des opérations linéaires. After the outbreak of World War II, in September 1939, Lwów was taken over by the Soviet Union. Banach became a member of the Academy of Sciences of Ukraine and was the dean of the Department of Mathematics of Physics of Lwów University. In 1941, when Germans took over the city, all institutions of higher education were closed to Poles. As a result, Banach had to earn money as a feeder of lice at Rudolf Weigl's Institute for Study of Typhus and Virology. While the job carried the risk of becoming infected with typhus, it protected him from being sent to slave labor in Germany and other forms of repression. When the Soviets recaptured Lwów in 1944, Banach reestablished the University. However, because the Soviets were removing Poles from annexed formerly Polish territories, Banach prepared to return to Kraków. He died in August 1945 after being diagnosed with lung cancer seven months earlier. Some of the notable mathematical concepts named after Banach include Banach spaces, Banach algebras, the Banach– Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach-Mazur game, the Banach– Alaoglu theorem and the Banach fixed-point theorem.

36.1 Life

36.1.1 Early life

Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in Kraków, then part of the Austro- Hungarian Empire. Banach’s parents were Stefan Greczek and Katarzyna Banach, both natives of the Podhale region.[1] Greczek was a soldier in the Austro-Hungarian Army stationed in Kraków. Little is known about Banach’s mother.[2] Unusually, Stefan’s surname was that of his mother instead of his father, though he received his father’s given name, Stefan. Since Stefan Greczek was a private and was prevented by military regulations from marrying, and the mother was too poor to support the child, the couple decided that he should be reared by family and friends.[3][4][5] Stefan spent the first few years of his life with his grandmother, but when she took ill Greczek arranged for his son to be raised by Franciszka Płowa and her niece Maria Puchalska in Kraków. Young Stefan would regard Franciszka as

144 36.1. LIFE 145 his foster mother and Maria as his older sister.[6] In his early years Banach was tutored by Juliusz Mien, a French intellectual and friend of the Płowa family, who had emigrated to Poland and supported himself with photography and translations of Polish literature into French. Mien taught Banach French and most likely encouraged him in his early mathematical pursuits.[4] In 1902 Banach, aged 10, enrolled in Kraków’s Henryk Sienkiewicz Gymnasium (also known as the Goetz Gymna- sium). While the school specialized in the humanities, Banach and his best friend Witold Wiłkosz (also a future mathematician) spent most of their time working on mathematics problems during breaks and after school.[7] Later in life Banach would credit Dr. Kamil Kraft, the mathematics and physics teacher at the gymnasium with kindling his interests in mathematics.[8] While generally Banach was a diligent student he did on occasion receive low grades (he failed Greek during his first semester at the gymnasium) and would later speak critically of the school’s math teachers.[5] After obtaining his matura (high school degree) at age 18 in 1910 Banach, together with Wiłkosz, moved to Lwów with the intention of studying at the Lwów Polytechnic. He initially chose engineering as his field of study since at the time he was convinced that there was nothing new to discover in mathematics.[9] At some point he also attended Jagiellonian University in Kraków on a part-time basis. As Banach had to earn money to support his studies it was not until 1914 that he finally, at age 22, passed his high school graduation exams.[10] When World War I broke out, Banach was excused from military service due to his left-handedness and poor vision. When the Russian Army opened its offensive toward Lwów, Banach left for Kraków, where he spent the rest of the war. He made his living as a tutor at the local gymnasiums, worked in a bookstore and as a foreman of road building crew. He may have attended lectures at the Jagiellonian University at that time, including those of the famous Polish mathematician Stanisław Zaremba, but little is known of that period of his life.[11]

36.1.2 Discovery by Steinhaus

In 1916, in Kraków’s Planty gardens, Banach encountered Professor Hugo Steinhaus, one of the renowned mathe- maticians of the time. According to Steinhaus, while he was strolling through the gardens he was surprised to overhear the term “Lebesgue measure” (Lebesgue integration was at the time still a fairly new idea in mathematics) and walked over to investigate. As a result he met Banach, as well as Otto Nikodym and Wilkosz.[12] Steinhaus became fascinated with the self-taught young mathematician. The encounter resulted in a long-lasting collaboration and friendship. In fact, soon after the encounter Steinhaus invited Banach to solve some problems he had been working on but which had proven difficult. Banach solved them within a week and the two soon published their first joint work (On the Mean Convergence of Fourier Series). Steinhaus, Banach and Nikodym, along with several other Kraków mathemati- cians (Władysław Ślebodziński, Leon Chwistek, Jan Kroć, and Włodzimierz Stożek) also established a mathematical society, which eventually became the Polish Mathematical Society.[13] The society was officially founded on April 2, 1919. It was also through Steinhaus that Banach met his future wife, Łucja Braus.

36.1.3 Interbellum

Steinhaus introduced Banach to academic and substantially accelerated his career. After Poland regained independence, in 1920 Banach was given an assistantship at Kraków’s Jagiellonian University. Steinhaus’ backing also allowed him to receive a doctorate without actually graduating from a university. The doctoral thesis, accepted by King John II Casimir University of Lwów in 1920 [14] and published in 1922,[15] included the basic ideas of functional analysis, which was soon to become an entirely new branch of mathematics. The thesis was widely discussed in academic circles and allowed him in 1922 to become a professor at the Lwów Polytechnic. Initially an assistant to Professor Antoni Łomnicki, in 1927 Banach received his own chair. In 1924 he was also accepted as a member of the Polish Academy of Learning. At the same time, from 1922, Banach also headed the second Chair of Mathematics at University of Lwów. Young and talented, Banach gathered around him a large group of mathematicians. The group, meeting in the Scottish Café, soon gave birth to the “Lwów School of Mathematics”. In 1929 the group began publishing its own journal, Studia Mathematica, devoted primarily to Banach’s field of study — functional analysis. Around that time, Banach also began working on his best-known work, the first monograph on the general theory of linear-metric space. First published in Polish in 1931,[16] the following year it was also translated into French and gained wider recognition in European academic circles.[17] The book was also the first in a long series of mathematics monographs edited by Banach and his circle. 146 CHAPTER 36. STEFAN BANACH

Scottish Café, meeting place of many famous Lwów mathematicians

36.1.4 World War II

Following the invasion of Poland by Nazi Germany and the Soviet Union, Lwów came under the control of the Soviet Union for almost two years. Banach, from 1939 a corresponding member of the Academy of Sciences of Ukraine, and on good terms with Soviet mathematicians,[2] had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities.[18] Following the German takeover of Lwów in 1941 during Operation Barbarossa, all universities were closed and Banach, along with many colleagues and his son, was employed as lice feeder at Professor Rudolf Weigl's Typhus Research Institute. Employment in Weigl’s Institute provided many unemployed university professors and their associates protection from random arrest and deportation to Nazi concentration camps. After the Red Army recaptured Lviv in the Lvov–Sandomierz Offensive of 1944, Banach returned to the University and helped re-establish it after the war years. However, because the Soviets were removing Poles from annexed formerly Polish territories, Banach began preparing to leave the city and settle in Kraków, Poland, where he had been promised a chair at the Jagiellonian University.[2] He was also considered a candidate for Minister of Education of Poland.[19] In January 1945, however, he was diagnosed with lung cancer and was allowed to stay in Lwów. He died on August 31, 1945, aged 53. His funeral at the Lychakiv Cemetery was attended by hundreds of people.[19]

36.2 Contributions

Banach’s dissertation, completed in 1920 and published in 1922, formally axiomatized the concept of a complete normed vector space and laid the foundations for the area of functional analysis. In this work Banach called such spaces “class E-spaces”, but in his 1932 book, Théorie des opérations linéaires, he changed terminology and referred to them as “spaces of type B”, which most likely contributed to the subsequent eponymous naming of these spaces after him.[20] The theory of what came to be known as Banach spaces had antecedents in the work of the Hungarian mathematician Frigyes Riesz (published in 1916) and contemporaneous contributions from Hans Hahn and Norbert Wiener.[14] For a brief period in fact, complete normed linear spaces were referred to as “Banach-Wiener” spaces in mathematical literature, based on terminology introduced by Wiener himself. However, because Wiener’s work on 36.2. CONTRIBUTIONS 147

Banach’s grave, Lychakiv Cemetery, Lviv (Lwów, in Polish)

Decomposition of a ball into two identical balls - the Banach–Tarski paradox. the topic was limited, the established name became just Banach spaces.[20] Likewise, Banach’s fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students (for example in the Banach–Schauder theorem) and other mathematicians (in particular Brouwer and Poincaré and Birkhoff). The theorem did not require linearity of the space, and applied to any Cauchy space ().[14] The Hahn–Banach theorem, is one of the fundamental theorems of functional analysis.[14]

• Banach–Tarski paradox

• Banach–Steinhaus theorem

• Banach–Alaoglu theorem

• Banach–Stone theorem 148 CHAPTER 36. STEFAN BANACH

36.3 Quotes

Banach monument, Kraków

Stanislaw Ulam, another mathematician of the Lwów School of Mathematics, in his autobiography, quotes Banach as saying:

“Good mathematicians see analogies. Great mathematicians see analogies between analogies.” 36.4. SEE ALSO 149

Hugo Steinhaus said of Banach:

“An exceptional intellect, exceptional discoveries... he gave Polish science... more than anybody else.” “Banach was my greatest scientific discovery.”

36.4 See also

• 16856 Banach

• Banach algebra

• Banach manifold

• Amenable Banach algebra

• Banach bundle

• Banach function algebra

•

• Banach’s matchbox problem

• Banach measure

• Closed range theorem

• Banach space

• List of Poles

36.5 Notes

[1] Waksmundzka-Hajnos 2006, p.16

[2] O'Connor and Robertson

[3] Kałuża 1996, p.2-3

[4] Kałuża 1996, p.3

[5] Kałuża 1996, p.3-4

[6] Kałuża 1996, p.1-3

[7] Kałuża 1996, p.137

[8] Jakimowicz & Miranowicz 2007, p. 4

[9] Jakimowicz & Miranowicz 2007, p.5

[10] Kałuża 1996, p.13

[11] Kałuża 1996, p.16

[12] Jakimowicz & Miranowicz 2007, p. 6

[13] Kałuża 1996, p. 23

[14] Jahnke 2003, p.402

[15] Stefan Banach (1922). “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”. Fun- damenta Mathematicae (in French and Polish) III.

[16] Stefan Banach: Teoria operacji liniowych. 150 CHAPTER 36. STEFAN BANACH

[17] Stefan Banach: Théorie des opérations linéaires (in French; Theory of Linear Operations).

[18] Urbaniak

[19] James 2003, p.384

[20] MacCluer 2008, p. 6

36.6 References

• Jahnke, Hans Niels (2003). A History of Analysis. American Mathematical Society. ISBN 0821826239. • Jakimowicz, E.; Miranowicz, A., eds. (2007). Stefan Banach - Remarkable life, Brilliant mathematics. Gdańsk University Press and Adam Mickiewicz University Press. ISBN 978-83-7326-451-9. • James, Ioan (2003). Remarkable Mathematicians: From Euler to von Neumann. Cambridge University Press. ISBN 0521520940. • Kałuża, Roman (1996). Through a Reporter’s Eyes: The Life of Stefan Banach. Translated by Wojbor Andrzej Woyczyński and Ann Kostant. Birkhäuser. ISBN 0-8176-3772-9. • Kosiedowski, Stanisław. “Stefan Banach”. Mój Lwów. Retrieved 2008-05-20.

• O'Connor, John J.; Robertson, Edmund F. (2000). “Stefan Banach”. MacTutor History of Mathematics archive. University of St. Andrews. Retrieved August 19, 2012.

• Siegmund-Schultze, Reinhard (2003). Jahnke, Hans Niels, ed. A History of Analysis. American Mathematical Society. ISBN 0-8218-2623-9.

• MacCluer, Barbara (2008). Elementary Functional Analysis. Springer. ISBN 0387855289. • Urbaniak, Mariusz (April 2002). “Geniusz: gen i już". Polityka 8 (2348).

• Waksmundzka-Hajnos, Monika (2006). “Wspomnienie o Stefanie Greczku”. Focus (Gdańsk University) (11).

36.7 External links

• Page devoted to Stefan Banach • Stefan Banach at the Mathematics Genealogy Project

• Works by or about Stefan Banach in libraries (WorldCat catalog) Chapter 37

Subset

“Superset” redirects here. For other uses, see Superset (disambiguation). In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

B A

Euler diagram showing A is a proper subset of B and conversely B is a proper superset of A

151 152 CHAPTER 37. SUBSET

“contained” inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. The subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

37.1 Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by A ⊆ B , or equivalently • B is a superset of (or includes) A, denoted by B ⊇ A.

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then

• A is also a proper (or strict) subset of B; this is written as A ⊊ B. or equivalently • B is a proper superset of A; this is written as B ⊋ A.

For any set S, the inclusion relation ⊆ is a partial order on the set P(S) of all subsets of S (the power set of S). When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.[1]

37.2 ⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and ⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x may not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.

37.3 Examples

• The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions A ⊆ B and A ⊊ B are true. • The set D = {1, 2, 3} is a subset of E = {1, 2, 3}, thus D ⊆ E is true, and D ⊊ E is not true (false). • Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.) • The empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any set except itself. • The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10} • The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition. • The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the latter set has a larger cardinality (or power) than the former set. 37.4. OTHER PROPERTIES OF INCLUSION 153

regular polygons

The regular polygons form a subset of the polygons

Another example in an :

• A is a proper subset of B • C is a subset but no proper subset of B

37.4 Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ⪯ ) is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b]. For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product of k = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating S S = {s1, s2, …, sk} and associating with each subset T ⊆ S (which is to say with each element of 2 ) the k-tuple from {0,1}k of which the ith coordinate is 1 if and only if si is a member of T.

37.5 See also

• Containment order

37.6 References

[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 924157 154 CHAPTER 37. SUBSET

C B A

A B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

37.7 External links

• Weisstein, Eric W., “Subset”, MathWorld. Chapter 38

Transitive set

In set theory, a set A is transitive, if and only if

• whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently, • whenever x ∈ A, and x is not an urelement, then x is a subset of A.

Similarly, a class M is transitive if every element of M is a subset of M.

38.1 Examples

Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). Any of the stages Vα and Lα leading to the construction of the von Neumann universe V and Gödel’s constructible universe L are transitive sets. The universes L and V themselves are transitive classes.

38.2 Properties ∪ ∪ ∪ A set X is transitive if and only if X ⊆ X∪ , where X is the union of all elements of X that are sets, X = {y | (∃x ∈ X)y ∈ x} . If X is transitive, then X is transitive. If X and Y are transitive,∪ then X∪Y∪{X,Y} is transitive. In general, if X is a class all of whose elements are transitive sets, then X ∪ X is transitive. A set X which does not contain urelements is transitive if and only if it is a subset of its own power set, X ⊂ P(X). The power set of a transitive set without urelements is transitive.

38.3 Transitive closure

The transitive closure of a set X is the smallest (with respect to inclusion) transitive set which contains X. Suppose one is given a set X, then the transitive closure of X is

∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ {X, X, X, X, X,...}. Note that this is the set of all of the objects related to X by the transitive closure of the membership relation.

38.4 Transitive models of set theory

Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models. The reason is that properties defined by bounded formulas are absolute for transitive classes.

155 156 CHAPTER 38. TRANSITIVE SET

A transitive set (or class) that is a model of a of set theory is called a transitive model of the system. Transitivity is an important factor in determining the absoluteness of formulas. In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity.[1]

38.5 See also

• End extension

• Transitive relation • Supertransitive class

38.6 References

[1] Goldblatt (1998) p.161

• Ciesielski, Krzysztof (1997), Set theory for the working mathematician, London Mathematical Society Student Texts 39, Cambridge: Cambridge University Press, ISBN 0-521-59441-3, Zbl 0938.03067 • Goldblatt, Robert (1998), Lectures on the hyperreals. An introduction to nonstandard analysis, Graduate Texts in Mathematics 188, New York, NY: Springer-Verlag, ISBN 0-387-98464-X, Zbl 0911.03032 • Jech, Thomas (2008) [originally published in 1973], The Axiom of Choice, Dover Publications, ISBN 0-486- 46624-8, Zbl 0259.02051

38.7 External links

• Weisstein, Eric W., “Transitive”, MathWorld.

• Weisstein, Eric W., “Transitive Closure”, MathWorld. • Weisstein, Eric W., “Transitive Reduction”, MathWorld. Chapter 39

Ultrafilter

In the mathematical field of set theory, an ultrafilter is a maximal filter, that is, a filter that cannot be enlarged. Filters and ultrafilters are special subsets of partially ordered sets. Ultrafilters can also be defined on Boolean algebras and sets:

• An ultrafilter on a poset P is a maximal filter on P.

• An ultrafilter on a Boolean algebra B is an ultrafilter on the poset of non-zero elements of B.

• An ultrafilter on a set X is an ultrafilter on the Boolean algebra of subsets of X.

Rather confusingly, an ultrafilter on a poset P or Boolean algebra B is a subset of P or B, while an ultrafilter on a set X is a collection of subsets of X. Ultrafilters have many applications in set theory, model theory, and topology. An ultrafilter on a set X has some special properties. For example, given any subset A of X, the ultrafilter must contain either A or its complement X \ A. In addition, an ultrafilter on a set X may be considered as a finitely additive measure. In this view, every subset of X is either considered "almost everything" (has measure 1) or “almost nothing” (has measure 0).

39.1 Formal definition for ultrafilter on a set

Given a set X, an ultrafilter on X is a set U consisting of subsets of X such that

1. The empty set is not an element of U

2. If A and B are subsets of X, A is a subset of B, and A is an element of U, then B is also an element of U.

3. If A and B are elements of U, then so is the intersection of A and B.

4. If A is a subset of X, then either A or X \ A is an element of U. (Note: axioms 1 and 3 imply that A and X \ A cannot both be elements of U.)

A characterization is given by the following theorem. A filter U on a set X is an ultrafilter if any of the following conditions are true:

1. There is no filter F finer than U, i.e., U ⊆ F implies U = F.

2. A ∪ B ∈ U implies A ∈ U or B ∈ U .

3. ∀A ⊆ X : A ∈ U or X \ A ∈ U .

Another way of looking at ultrafilters on a set X is to define a function m on the power set of X by setting m(A) = 1 if A is an element of U and m(A) = 0 otherwise. Such a function is called a 2-valued morphism. Then m is a

157 158 CHAPTER 39.

finitely additive measure on X, and every property of elements of X is either true almost everywhere or false almost everywhere. Note that this does not define a measure in the usual sense, which is required to be countably additive. For a filter F that is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefined elsewhere. A simple example of an ultrafilter is a principal ultrafilter, which consists of subsets of X that contain a given element x of X. All ultrafilters on a finite set are principal.

39.2 Completeness

The completeness of an ultrafilter U on a set is the smallest cardinal κ such that there are κ elements of U whose intersection is not in U. The definition implies that the completeness of any ultrafilter is at least ℵ0 . An ultrafilter whose completeness is greater than ℵ0 —that is, the intersection of any countable collection of elements of U is still in U—is called countably complete or σ -complete. The completeness of a countably complete nonprincipal ultrafilter on a set is always a measurable cardinal.

39.3 Generalization to partial orders

In order theory, an ultrafilter is a subset of a partially ordered set (a poset) that is maximal among all proper filters. Formally, this states that any filter that properly contains an ultrafilter has to be equal to the whole poset.

39.4 Special case: Boolean algebra

An important special case of the concept occurs if the considered poset is a Boolean algebra, as in the case of an ultrafilter on a set (defined as a filter of the corresponding powerset). In this case, ultrafilters are characterized by containing, for each element a of the Boolean algebra, exactly one of the elements a and ¬a (the latter being the Boolean complement of a). Ultrafilters on a Boolean algebra can be identified with prime ideals, maximal ideals, and homomorphisms to the 2-element Boolean algebra {true, false}, as follows:

• Maximal ideals of a Boolean algebra are the same as prime ideals. • Given a homomorphism of a Boolean algebra onto {true, false}, the inverse image of “true” is an ultrafilter, and the inverse image of “false” is a maximal ideal. • Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomor- phism onto {true, false} taking the maximal ideal to “false”. • Given an ultrafilter of a Boolean algebra, its complement is a maximal ideal, and there is a unique homomor- phism onto {true, false} taking the ultrafilter to “true”.

Let us see another theorem, which could be used for the definition of the concept of “ultrafilter”. Let B denote a Boolean algebra and F a proper filter[1] in it. F is an ultrafilter iff:

for all a, b ∈ B , if a ∨ b ∈ F , then a ∈ F or b ∈ F

(To avoid confusion: the ∨ denotes the join operation of the Boolean algebra, and logical connectives are rendered by English circumlocutions.) See details (and proof) in.[2]

39.5 Types and existence of ultrafilters

There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form Fa = {x | a ≤ x} for some (but 39.6. APPLICATIONS 159

not all) elements a of the given poset. In this case a is called the principal element of the ultrafilter. For the case of ultrafilters on sets, the elements that qualify as principals are exactly the one-element sets. Thus, a principal ultrafilter on a set S consists of all sets containing a particular point of S. An ultrafilter on a finite set is principal. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter. Note that an ultrafilter on an infinite set S is non-principal if and only if it contains the Fréchet filter of cofinite subsets of S. This is obvious, since a non-principal ultrafilter contains no finite set, it means that, by taking complements, it contains all cofinite subsets of S, which is exactly the Fréchet filter. One can show that every filter of a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see Ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice (AC) in the form of Zorn’s Lemma. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). Proofs involving the axiom of choice do not produce explicit examples of free ultrafilters. Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter of a finite poset (or on a finite set) is principal, since any finite filter has a least element. In ZF without the axiom of choice, it is possible that every ultrafilter is principal.

39.6 Applications

Ultrafilters on sets are useful in topology, especially in relation to compact Hausdorff spaces, and in model theory in the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on posets are most important if the poset is a Boolean algebra, since in this case the ultrafilters coincide with the prime filters. Ultrafilters in this form play a central role in Stone’s representation theorem for Boolean algebras. The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element a of P, let Da = {U ∈ G | a ∈ U}. This is most useful when P is again a Boolean algebra, since in this situation the set of all Da is a base for a compact Hausdorff topology on G. Especially, when considering the ultrafilters on a set S (i.e., the case that P is the powerset of S ordered via subset inclusion), the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality |S|. The ultraproduct construction in model theory uses ultrafilters to produce elementary extensions of structures. For example, in constructing hyperreal numbers as an ultraproduct of the real numbers, we first extend the domain of discourse from the real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead we define the functions and relations “pointwise modulo U", where U is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic. If U is nonprincipal, then the extension thereby obtained is nontrivial. In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. This con- struction yields a rigorous way to consider looking at the group from infinity, that is the large scale geometry of the group. Asymptotic cones are particular examples of ultralimits of metric spaces. Gödel’s ontological proof of God’s existence uses as an axiom that the set of all “positive properties” is an ultrafilter. In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for ag- gregating the preferences of infinitely many individuals. Contrary to Arrow’s impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (e.g., Kirman and Sondermann, 1972[3]). Mihara (1997,[4] 1999[5]) shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable. 160 CHAPTER 39. ULTRAFILTER

39.7 Ordering on ultrafilters

Rudin–Keisler ordering is a preorder on the class of ultrafilters defined as follows: if U is an ultrafilter on X, and V an ultrafilter on Y, then V ≤RK U if and only if there exists a function f: X → Y such that

C ∈ V ⇐⇒ f −1[C] ∈ U

for every subset C of Y.

Ultrafilters U and V are Rudin–Keisler equivalent, U ≡RK V , if there exist sets A ∈ U , B ∈ V , and a bijection f: A → B that satisfies the condition above. (If X and Y have the same cardinality, the definition can be simplified by fixing A = X, B = Y.)

It is known that ≡RK is the kernel of ≤RK , i.e., U ≡RK V if and only if U ≤RK V and V ≤RK U .

39.8 Ultrafilters on ω

There are several special properties that an ultrafilter on ω may possess, which prove useful in various areas of set theory and topology.

• A non-principal ultrafilter U is a P-point (or weakly selective) iff for every partition of ω, {Cn | n < ω} such that Cn ̸∈ U, ∀n < ω , there exists A ∈ U such that |A ∩ Cn| < ω, ∀n < ω .

• A non-principal ultrafilter U is Ramsey (or selective) iff for every partition of ω, {Cn | n < ω} such that Cn ̸∈ U, ∀n < ω , there exists A ∈ U such that |A ∩ Cn| = 1, ∀n < ω

It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters.[6] In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin’s axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters.[7] Therefore the existence of these types of ultrafilters is independent of ZFC. P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non- principal ultrafilters. The name Ramsey comes from Ramsey’s theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of [ω]2 there exists an element of the ultrafilter that has a homogeneous color. An ultrafilter on ω is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal ultrafilters.

39.9 See also

• Universal net

39.10 Notes

[1] I.e., a filter F with the surplus restriction 0∈ / F , i.e., being a filter that does not “degenerate” to coincide with the whole (universe of the) Boolean algebra

[2] A Course in Universal Algebra (written by Stanley N. Burris and H.P. Sankappanavar), Corollary 3.13 on p. 149.

[3] Kirman, A.; Sondermann, D. (1972). “Arrow’s theorem, many agents, and invisible dictators”. Journal of Economic Theory 5: 267. doi:10.1016/0022-0531(72)90106-8.

[4] Mihara, H. R. (1997). “Arrow’s Theorem and Turing computability” (PDF). Economic Theory 10 (2): 257–276. doi:10.1007/s001990050157Reprinted in K. V. Velupillai , S. Zambelli, and S. Kinsella, ed., Computable Economics, International Library of Critical Writings in Economics, Edward Elgar, 2011.

[5] Mihara, H. R. (1999). “Arrow’s theorem, countably many agents, and more visible invisible dictators”. Journal of Mathe- matical Economics 32: 267–277. doi:10.1016/S0304-4068(98)00061-5. 39.11. REFERENCES 161

[6] Rudin, Walter (1956), “Homogeneity problems in the theory of Čech compactifications”, Duke Mathematical Journal 23 (3): 409–419, doi:10.1215/S0012-7094-56-02337-7

[7] Wimmers, Edward (March 1982), “The Shelah P-point independence theorem”, Israel Journal of Mathematics (Hebrew University Magnes Press) 43 (1): 28–48, doi:10.1007/BF02761683

39.11 References

• Comfort, W. W. (1977), “Ultrafilters: some old and some new results”, Bulletin of the American Mathematical Society 83 (4): 417–455, doi:10.1090/S0002-9904-1977-14316-4, ISSN 0002-9904, MR 0454893

• Comfort, W. W.; Negrepontis, S. (1974), The theory of ultrafilters, Berlin, New York: Springer-Verlag, MR 0396267 Chapter 40

Ultraproduct

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elemen- tary equivalence, and the Robinson-Zakon presentation of the use of superstructures and their to construct nonstandard models of analysis, leading to the growth of the area of non-standard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.

40.1 Definition

The general method for getting ultraproducts uses an index set I, a structure Mi for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be infinite and U to contain all cofinite subsets of I. Otherwise the ultrafilter is principal, and the ultraproduct is isomorphic to one of the factors. Algebraic operations on the Cartesian product

∏ Mi i∈I are defined in the usual way (for example, for a binary function +, (a + b) i = ai + bi ), and an equivalence relation is defined by a ~ b if and only if

{i ∈ I : ai = bi} ∈ U, and the ultraproduct is the quotient set with respect to ~. The ultraproduct is therefore sometimes denoted by

∏ Mi/U. i∈I One may define a finitely additive measure m on the index set I by saying m(A) = 1 if A ∈ U and = 0 otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set. The ultraproduct is the set of equivalence classes thus generated. Other relations can be extended the same way:

{ } 1 n ⇐⇒ ∈ Mi 1 n ∈ R([a ],..., [a ]) i I : R (ai , . . . , ai ) U,

162 40.2. EXAMPLES 163

where [a] denotes the equivalence class of a with respect to ~. In particular, if every Mi is an ordered field, then so is the ultraproduct. An ultrapower is an ultraproduct for which all the factors Mi are equal:

∏ M κ/U = M/U. α<κ ∏ More generally, the construction above can be carried out whenever U is a filter on I; the resulting model i∈I Mi/U is then called a reduced product.

40.2 Examples

The hyperreal numbers are the ultraproduct of one copy of the real numbers for every natural number, with regard to an ultrafilter over the natural numbers containing all cofinite sets. Their order is the extension of the order of the real numbers. For example, the sequence ω given by ωi = i defines an equivalence class representing a hyperreal number that is greater than any real number. Analogously, one can define nonstandard integers, nonstandard complex numbers, etc., by taking the ultraproduct of copies of the corresponding structures. As an example of the carrying over of relations into the ultraproduct, consider the sequence ψ defined by ψi = 2i. Because ψi > ωi = i for all i, it follows that the equivalence class of ψi = 2i is greater than the equivalence class of ωi = i, so that it can be interpreted as an infinite number which is greater than the one originally constructed. However, let χi = i for i not equal to 7, but χ7 = 8. The set of indices on which ω and χ agree is a member of any ultrafilter (because ω and χ agree almost everywhere), so ω and χ belong to the same equivalence class. In the theory of large cardinals, a standard construction is to take the ultraproduct of the whole set-theoretic universe with respect to some carefully chosen ultrafilter U. Properties of this ultrafilter U have a strong influence on (higher order) properties of the ultraproduct; for example, if U is σ-complete, then the ultraproduct will again be well- founded. (See measurable cardinal for the prototypical example.)

40.3 Łoś's theorem

Łoś's theorem, also called the fundamental theorem of ultraproducts, is due to Jerzy Łoś (the surname is pronounced [ˈwɔɕ], approximately “wash”). It states that any first-order formula is true in the ultraproduct if and only if the set of indices i such that the formula is true in Mi is a member of U. More precisely: ∈ Let σ be a signature, U an ultrafilter over a set∏I , and for each i I let Mi be a σ-structure.∏ Let M be the ultraproduct 1 n ∈ k k of the Mi with respect to U , that is, M = i∈I Mi/U. Then, for each a , . . . , a Mi , where a = (ai )i∈I , and for every σ-formula ϕ ,

| 1 n ⇐⇒ { ∈ | 1 n } ∈ M = ϕ[[a ],..., [a ]] i I : Mi = ϕ[ai , . . . , ai ] U.

The theorem is proved by induction on the complexity of the formula ϕ . The fact that U is an ultrafilter (and not just a filter) is used in the negation clause, and the axiom of choice is needed at the existential quantifier step. As an application, one obtains the transfer theorem for hyperreal fields.

40.3.1 Examples

Let R be a unary relation in the structure M, and form the ultrapower of M. Then the set S = {x ∈ M|Rx} has an analog *S in the ultrapower, and first-order formulas involving S are also valid for *S. For example, let M be the reals, and let Rx hold if x is a . Then in M we can say that for any pair of rationals x and y, there exists another number z such that z is not rational, and x < z < y. Since this can be translated into a first-order logical formula in the relevant formal language, Łoś's theorem implies that *S has the same property. That is, we can define a notion 164 CHAPTER 40. ULTRAPRODUCT of the hyperrational numbers, which are a subset of the hyperreals, and they have the same first-order properties as the rationals. Consider, however, the Archimedean property of the reals, which states that there is no real number x such that x > 1, x > 1 +1 , x > 1 + 1 + 1, ... for every inequality in the infinite list. Łoś's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic. In fact, the Archimedean property is false for the hyperreals, as shown by the construction of the hyperreal number ω above.

40.4 Ultralimit

For the ultraproduct of a sequence of metric spaces, see Ultralimit.

In model theory and set theory, an ultralimit or limiting ultrapower is a direct limit of a sequence of ultrapowers.

Beginning with a structure, A0, and an ultrafilter, D0, form an ultrapower, A1. Then repeat the process to form A2, and so forth. For each n there is a canonical diagonal embedding An → An+1 . At limit stages, such as Aω, form the direct limit of earlier stages. One may continue into the transfinite.

40.5 References

• Bell, John Lane; Slomson, Alan B. (2006) [1969]. Models and Ultraproducts: An Introduction (reprint of 1974 edition ed.). Dover Publications. ISBN 0-486-44979-3.

• Burris, Stanley N.; Sankappanavar, H.P. (2000) [1981]. A Course in Universal Algebra (Millennium edition ed.). Chapter 41

Uncountable set

“Uncountable” redirects here. For the linguistic concept, see Uncountable noun.

In mathematics, an uncountable set (or uncountably infinite set)[1] is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

41.1 Characterizations

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds:

• There is no injective function from X to the set of natural numbers.

• X is nonempty and every ω-sequence of elements of X fails to include at least one element of X. That is, X is nonempty and there is no surjective function from the natural numbers to X.

• The cardinality of X is neither finite nor equal to ℵ0 (aleph-null, the cardinality of the natural numbers).

• The set X has cardinality strictly greater than ℵ0 .

The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

41.2 Properties

• If an uncountable set X is a subset of set Y, then Y is uncountable.

41.3 Examples

The best known example of an uncountable set is the set R of all real numbers; Cantor’s diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural ℵ0 numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or 2 , or ℶ1 (beth-one). The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable.

165 166 CHAPTER 41. UNCOUNTABLE SET

Another example of an uncountable set is the set of all functions from R to R. This set is even “more uncountable” than R in the sense that the cardinality of this set is ℶ2 (beth-two), which is larger than ℶ1 .

A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω1. The cardinality of Ω is denoted ℵ1 (aleph-one). It can be shown, using the axiom of choice, that ℵ1 is the smallest uncountable cardinal number. Thus either ℶ1 , the cardinality of the reals, is equal to ℵ1 or it is strictly larger. Georg Cantor was the first to propose the question of whether ℶ1 is equal to ℵ1 . In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that ℵ1 = ℶ1 is now called the continuum hypothesis and is known to be independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice).

41.4 Without the axiom of choice

Without the axiom of choice, there might exist cardinalities incomparable to ℵ0 (namely, the cardinalities of Dedekind- finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth charac- terization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable. If the axiom of choice holds, the following conditions on a cardinal κare equivalent:

• κ ≰ ℵ0;

• κ > ℵ0; and

• κ ≥ ℵ1 , where ℵ1 = |ω1| and ω1 is least initial ordinal greater than ω.

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of “uncountability” when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.

41.5 See also

• Aleph number

• Beth number • Injective function

• Natural number

41.6 References

[1] Uncountably Infinite — from Wolfram MathWorld

• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer- Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition). • Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN 3-540-44085-2

41.7 External links

• Proof that R is uncountable Chapter 42

Universe (mathematics)

In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains (as elements) all the entities one wishes to consider in a given situation. There are several versions of this general idea, described in the following sections.

42.1 In a specific context

Perhaps the simplest version is that any set can be a universe, so long as the object of study is confined to that particular set. If the object of study is formed by the real numbers, then the real line R, which is the real number set, could be the universe under consideration. Implicitly, this is the universe that Georg Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis. The only sets that Cantor was originally interested in were subsets of R. This concept of a universe is reflected in the use of Venn diagrams. In a Venn diagram, the action traditionally takes place inside a large rectangle that represents the universe U. One generally says that sets are represented by circles; but these sets can only be subsets of U. The complement of a set A is then given by that portion of the rectangle outside of A's circle. Strictly speaking, this is the relative complement U \ A of A relative to U; but in a context where U is the universe, it can be regarded as the absolute complement AC of A. Similarly, there is a notion of the nullary intersection, that is the intersection of zero sets (meaning no sets, not null sets). Without a universe, the nullary intersection would be the set of absolutely everything, which is generally regarded as impossible; but with the universe in mind, the nullary intersection can be treated as the set of everything under consideration, which is simply U. These conventions are quite useful in the algebraic approach to basic set theory, based on Boolean lattices. Except in some non-standard forms of axiomatic set theory (such as New Foundations), the class of all sets is not a Boolean lattice (it is only a relatively complemented lattice). In contrast, the class of all subsets of U, called the power set of U, is a Boolean lattice. The absolute complement described above is the complement operation in the Boolean lattice; and U, as the nullary intersection, serves as the top element (or nullary meet) in the Boolean lattice. Then De Morgan’s laws, which deal with complements of meets and joins (which are unions in set theory) apply, and apply even to the nullary meet and the nullary join (which is the empty set).

42.2 In ordinary mathematics

However, once subsets of a given set X (in Cantor’s case, X = R) are considered, the universe may need to be a set of subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will not themselves be subsets of X but will instead be subsets of PX, the power set of X. This may be continued; the object of study may next consist of such sets of subsets of X, and so on, in which case the universe will be P(PX). In another direction, the binary relations on X (subsets of the Cartesian product X × X) may be considered, or functions from X to itself, requiring universes like P(X × X) or XX. Thus, even if the primary interest is X, the universe may need to be considerably larger than X. Following the above ideas, one may want the superstructure over X as the universe. This can be defined by structural recursion as follows:

167 168 CHAPTER 42. UNIVERSE (MATHEMATICS)

• Let S0X be X itself.

• Let S1X be the union of X and PX.

• Let S2X be the union of S1X and P(S1X).

• In general, let Sn₊₁X be the union of SX and P(SnX).

Then the superstructure over X, written SX, is the union of S0X, S1X, S2X, and so on; or

∪∞ SX := SiX. i=0

Note that no matter what set X is the starting point, the empty set {} will belong to S1X. The empty set is the von Neumann ordinal [0]. Then {[0]}, the set whose only element is the empty set, will belong to S2X; this is the von Neumann ordinal [1]. Similarly, {[1]} will belong to S3X, and thus so will {[0],[1]}, as the union of {[0]} and {[1]}; this is the von Neumann ordinal [2]. Continuing this process, every natural number is represented in the superstructure by its von Neumann ordinal. Next, if x and y belong to the superstructure, then so does {{x},{x,y}}, which represents the ordered pair (x,y). Thus the superstructure will contain the various desired Cartesian products. Then the superstructure also contains functions and relations, since these may be represented as subsets of Cartesian products. The process also gives ordered n-tuples, represented as functions whose domain is the von Neumann ordinal [n]. And so on. So if the starting point is just X = {}, a great deal of the sets needed for mathematics appear as elements of the superstructure over {}. But each of the elements of S{} will be finite sets! Each of the natural numbers belongs to it, but the set N of all natural numbers does not (although it is a subset of S{}). In fact, the superstructure over {} consists of all of the hereditarily finite sets. As such, it can be considered the universe of finitist mathematics. Speaking anachronistically, one could suggest that the 19th-century finitist Leopold Kronecker was working in this universe; he believed that each natural number existed but that the set N (a "completed infinity") did not. However, S{} is unsatisfactory for ordinary mathematicians (who are not finitists), because even though N may be available as a subset of S{}, still the power set of N is not. In particular, arbitrary sets of real numbers are not available. So it may be necessary to start the process all over again and form S(S{}). However, to keep things simple, one can take the set N of natural numbers as given and form SN, the superstructure over N. This is often considered the universe of ordinary mathematics. The idea is that all of the mathematics that is ordinarily studied refers to elements of this universe. For example, any of the usual constructions of the real numbers (say by Dedekind cuts) belongs to SN. Even non-standard analysis can be done in the superstructure over a non-standard model of the natural numbers. One should note a slight shift in philosophy from the previous section, where the universe was any set U of interest. There, the sets being studied were subsets of the universe; now, they are members of the universe. Thus although P(SX) is a Boolean lattice, what is relevant is that SX itself is not. Consequently, it is rare to apply the notions of Boolean lattices and Venn diagrams directly to the superstructure universe as they were to the power-set universes of the previous section. Instead, one can work with the individual Boolean lattices PA, where A is any relevant set belonging to SX; then PA is a subset of SX (and in fact belongs to SX). In Cantor’s case X = R in particular, arbitrary sets of real numbers are not available, so there it may indeed be necessary to start the process all over again.

42.3 In set theory

It is possible to give a precise meaning to the claim that SN is the universe of ordinary mathematics; it is a model of Zermelo set theory, the axiomatic set theory originally developed by Ernst Zermelo in 1908. Zermelo set theory was successful precisely because it was capable of axiomatising “ordinary” mathematics, fulfilling the programme begun by Cantor over 30 years earlier. But Zermelo set theory proved insufficient for the further development of axiomatic set theory and other work in the foundations of mathematics, especially model theory. For a dramatic example, the description of the superstructure process above cannot itself be carried out in Zermelo set theory! The final step, forming S as an infinitary union, requires the axiom of replacement, which was added to Zermelo set theory in 1922 to form Zermelo–Fraenkel set theory, the set of axioms most widely accepted today. So while ordinary mathematics may be done in SN, discussion of SN goes beyond the “ordinary”, into metamathematics. 42.4. IN CATEGORY THEORY 169

But if high-powered set theory is brought in, the superstructure process above reveals itself to be merely the beginning of a transfinite recursion. Going back to X = {}, the empty set, and introducing the (standard) notation Vi for Si{}, V0 = {}, V1 = P{}, and so on as before. But what used to be called “superstructure” is now just the next item on the list: Vω, where ω is the first infinite ordinal number. This can be extended to arbitrary ordinal numbers:

∪ Vi := PVj j

∪ V := Vi i

Note that every individual Vi is a set, but their union V is a proper class. The axiom of foundation, which was added to ZF set theory at around the same time as the axiom of replacement, says that every set belongs to V.

Kurt Gödel's constructible universe L and the axiom of constructibility Inaccessible cardinals yield models of ZF and sometimes additional axioms, and are equivalent to the existence of the Grothendieck universe set

42.4 In category theory

There is another approach to universes which is historically connected with category theory. This is the idea of a Grothendieck universe. Roughly speaking, a Grothendieck universe is a set inside which all the usual operations of set theory can be performed. This version of a universe is defined to be any set for which the following axioms hold: [1]

1. x ∈ u ∈ U implies x ∈ U

2. u ∈ U and v ∈ U imply {u,v}, (u,v), and u × v ∈ U .

3. x ∈ U implies Px ∈ U and ∪x ∈ U

4. ω ∈ U (here ω = {0, 1, 2, ...} is the set of all finite ordinals.)

5. if f : a → b is a surjective function with a ∈ U and b ⊂ U , then b ∈ U .

The advantage of a Grothendieck universe is that it is actually a set, and never a proper class. The disadvantage is that if one tries hard enough, one can leave a Grothendieck universe. The most common use of a Grothendieck universe U is to take U as a replacement for the category of all sets. One says that a set S is U-small if S ∈U, and U-large otherwise. The category U-Set of all U-small sets has as objects all U-small sets and as morphisms all functions between these sets. Both the object set and the morphism set are sets, so it becomes possible to discuss the category of “all” sets without invoking proper classes. Then it becomes possible to define other categories in terms of this new category. For example, the category of all U-small categories is the category of all categories whose object set and whose morphism set are in U. Then the usual arguments of set theory are applicable to the category of all categories, and one does not have to worry about accidentally talking about proper classes. Because Grothendieck universes are extremely large, this suffices in almost all applications. Often when working with Grothendieck universes, mathematicians assume the Axiom of Universes: “For any set x, there exists a universe U such that x ∈U.” The point of this axiom is that any set one encounters is then U-small for some U, so any argument done in a general Grothendieck universe can be applied. This axiom is closely related to the existence of strongly inaccessible cardinals.

Set-like toposes 170 CHAPTER 42. UNIVERSE (MATHEMATICS)

42.5 See also

• Herbrand universe

• Free object

42.6 Notes

[1] Mac Lane 1998, p.22

42.7 References

• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag New York, Inc.

42.8 External links

• Hazewinkel, Michiel, ed. (2001), “Universe”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4 • Weisstein, Eric W., “Universal Set”, MathWorld. Chapter 43

Well-founded relation

“Noetherian induction” redirects here. For the use in topology, see Noetherian topological space.

In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-empty subset SX has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is not smaller than”) for the rest of the s ∈ S.

∀S ⊆ X (S ≠ ∅ → ∃m ∈ S ∀s ∈ S (s, m)/∈ R)

(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.) Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descend- ing chains: that is, there is no infinite sequence x0, x1, x2, ... of elements of X such that xn₊₁ R x for every natural number n. In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order. In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded. A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R−1 is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewriting systems, a Noetherian relation is also called terminating.

43.1 Induction and recursion

An important reason that well-founded relations are interesting is because a version of transfinite induction can be used on them: if (X, R) is a well-founded relation, P(x) is some property of elements of X, and we want to show that

P(x) holds for all elements x of X, it suffices to show that:

If x is an element of X and P(y) is true for all y such that y R x, then P(x) must also be true.

That is,

∀x ∈ X [(∀y ∈ X (y R x → P (y))) → P (x)] → ∀x ∈ XP (x).

171 172 CHAPTER 43. WELL-FOUNDED RELATION

Well-founded induction is sometimes called Noetherian induction,[1] after Emmy Noether. On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X, R) be a set-like well-founded relation and F a function that assigns an object F(x, g) to each pair of an element x ∈ X and a function g on the initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

G(x) = F (x, G|{y:y R x})

That is, if we want to construct a function G on X, we may define G(x) using the values of G(y) for y R x. As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graph of the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The statement that (N, <) is well-founded is also known as the well-ordering principle. There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction. When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those articles for more details.

43.2 Examples

Well-founded relations which are not totally ordered include:

• the positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if a divides b and a ≠ b. • the set of all finite strings over a fixed alphabet, with the order defined by s < t if and only if s is a proper substring of t.

• the set N × N of pairs of natural numbers, ordered by (n1, n2) < (m1, m2) if and only if n1 < m1 and n2 < m2. • the set of all regular expressions over a fixed alphabet, with the order defined by s < t if and only if s is a proper subexpression of t. • any class whose elements are sets, with the relation ∈ (“is an element of”). This is the axiom of regularity. • the nodes of any finite directed acyclic graph, with the relation R defined such that a R b if and only if there is an edge from a to b.

Examples of relations that are not well-founded include:

• the negative integers {−1, −2, −3, …}, with the usual order, since any unbounded subset has no least element. • The set of strings over a finite alphabet with more than one element, under the usual (lexicographic) order, since the sequence “B” > “AB” > “AAB” > “AAAB” > … is an infinite descending chain. This relation fails to be well-founded even though the entire set has a minimum element, namely the empty string. • the rational numbers (or reals) under the standard ordering, since, for example, the set of positive rationals (or reals) lacks a minimum.

43.3 Other properties

If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n − 1, n − 2, ..., 2, 1 has length n for any n. 43.4. REFLEXIVITY 173

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded rela- tions: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R) is isomorphic to (C,∈).

43.4 Reflexivity

A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example, in the natural numbers with their usual order ≤, we have 1 ≥ 1 ≥ 1 ≥ · · · . To avoid these trivial descending sequences, when working with a reflexive relation R it is common to use (perhaps implicitly) the alternate relation R′ defined such that a R′ b if and only if a R b and a ≠ b. In the context of the natural numbers, this means that the relation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of a well-founded relation is changed from the definition above to include this convention.

43.5 References

[1] Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley.

• Just, Winfried and Weese, Martin, Discovering Modern Set theory. I, American Mathematical Society (1998) ISBN 0-8218-0266-6. Chapter 44

Zermelo–Fraenkel set theory

“ZFC” redirects here. For other uses, see ZFC (disambiguation).

In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zer- melo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell’s paradox. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. ZFC is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of ZFC refer only to sets, not to urelements (elements of sets that are not themselves sets) or classes (collections of mathematical objects defined by a property shared by their members). The axioms of ZFC prevent its models from containing urelements, and proper classes can only be treated indirectly. Formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted ∈. The formula a ∈ b means that the set a is a member of the set b (which is also read, "a is an element of b" or "a is in b"). There are many equivalent formulations of the ZFC axioms. Most of the ZFC axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets a and b there is a new set {a, b} containing exactly a and b. Other axioms describe properties of set membership. A goal of the ZFC axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). The metamathematics of ZFC has been extensively studied. Landmark results in this area established the indepen- dence of the continuum hypothesis from ZFC, and of the axiom of choice from the remaining ZFC axioms. The consistency of a theory such as ZFC cannot be proved within the theory itself.

44.1 History

In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. However, as first pointed out by Abraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of the time, notably, the cardinal number ℵω and, where Z0 is any infinite set and ℘ is the power set operation, the set {Z0, ℘(Z0), ℘(℘(Z0)),...} (Ebbinghaus 2007, p. 136). Moreover, one of Zermelo’s axioms invoked a concept, that of a “definite” property, whose operational meaning was not clear. In 1922, Fraenkel and Thoralf Skolem independently proposed operationalizing a “definite” property as one that could be formulated as a first order theory whose atomic formulas were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed by Dimitry Mirimanoff in 1917), to Zermelo set theory yields the theory denoted by ZF. Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.

174 44.2. AXIOMS 175

44.2 Axioms

There are many equivalent formulations of the ZFC axioms; for a rich but somewhat dated discussion of this fact, see Fraenkel et al. (1973). The following particular axiom set is from Kunen (1980). The axioms per se are expressed in the symbolism of first order logic. The associated English prose is only intended to aid the intuition. All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below (although he notes that he does so only “for emphasis” (ibid., p. 10)). Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself, ∃x(x=x). Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC there are only sets, the interpretation of this logical theorem in the context of ZFC is that some set exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called free logic, in which it is not provable from logic alone that something exists, the axiom of infinity (below) asserts that an infinite set exists. This implies that a set exists and so, once again, it is superfluous to include an axiom asserting as much.

44.2.1 1.

Main article: Axiom of extensionality

Two sets are equal (are the same set) if they have the same elements.

∀x∀y[∀z(z ∈ x ⇔ z ∈ y) ⇒ x = y].

The converse of this axiom follows from the substitution property of equality. If the background logic does not include equality "=", x=y may be defined as an abbreviation for the following formula (Hatcher 1982, p. 138, def. 1):

∀z[z ∈ x ⇔ z ∈ y] ∧ ∀w[x ∈ w ⇔ y ∈ w].

In this case, the axiom of extensionality can be reformulated as

∀x∀y[∀z(z ∈ x ⇔ z ∈ y) ⇒ ∀w(x ∈ w ⇔ y ∈ w)], which says that if x and y have the same elements, then they belong to the same sets. (Fraenkel et al. 1973)

44.2.2 2. Axiom of regularity (also called the Axiom of foundation)

Main article: Axiom of regularity

Every non-empty set x contains a member y such that x and y are disjoint sets.

∀x[∃a(a ∈ x) ⇒ ∃y(y ∈ x ∧ ¬∃z(z ∈ y ∧ z ∈ x))].

This implies, for example, that no set is an element of itself and that every set has an ordinal rank.

44.2.3 3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension)

Main article: Axiom schema of specification 176 CHAPTER 44. ZERMELO–FRAENKEL SET THEORY

Subsets are commonly constructed using set builder notation. For example, the even integers can be constructed as the subset of the integers Z satisfying the predicate x ≡ 0 (mod 2) :

{x ∈ Z : x ≡ 0 (mod 2)}.

In general, the subset of a set z obeying a formula ϕ (x) with one free variable x may be written as:

{x ∈ z : ϕ(x)}.

The axiom schema of specification states that this subset always exists (it is an axiom schema because there is one ax- iom for each ϕ ). Formally, let ϕbe any formula in the language of ZFC with all free variables among x, z, w1, . . . , wn (y is not free in ϕ). Then:

∀z∀w1∀w2 ... ∀wn∃y∀x[x ∈ y ⇔ (x ∈ z ∧ ϕ)].

Note that the axiom schema of specification can only construct subsets, and does not allow the construction of sets of the more general form:

{x : ϕ(x)}.

This restriction is necessary to avoid Russell’s paradox and its variants that accompany naive set theory with unrestricted comprehension. In some other axiomatizations of ZF, this axiom is redundant in that it follows from the axiom schema of replacement. The axiom of specification can be used to prove the existence of the empty set, denoted ∅ , once at least one set is known to exist (see above). One way to do this is to use a property ϕ which no set has. For example, if w is any existing set, the empty set can be constructed as

∅ = {u ∈ w | (u ∈ u) ∧ ¬(u ∈ u)}

Thus the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on w). It is common to make a definitional extension that adds the symbol ∅ to the language of ZFC.

44.2.4 4. Axiom of pairing

Main article: Axiom of pairing

If x and y are sets, then there exists a set which contains x and y as elements.

∀x∀y∃z(x ∈ z ∧ y ∈ z).

The axiom schema of specification must be used to reduce this to a set with exactly these two elements. This axiom is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement, if we are given a set with at least two elements. The existence of a set with at least two elements is assured by either the axiom of infinity, or by the axiom schema of specification and the axiom of the power set applied twice to any set.

44.2.5 5. Axiom of union

Main article: Axiom of union 44.2. AXIOMS 177

The union over the elements of a set exists. For example, the union over the elements of the set {{1, 2}, {2, 3}} is {1, 2, 3} . Formally, for any set F there is a set A containing every element that is a member of some member of F :

∀F ∃A ∀Y ∀x[(x ∈ Y ∧ Y ∈ F) ⇒ x ∈ A].

f(x) x A B f : A → B

Axiom schema of replacement: the image of the domain set A under the definable function f (i.e. the range of f) falls inside a set B.

44.2.6 6. Axiom schema of replacement

Main article: Axiom schema of replacement

The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set.

Formally, let ϕbe any formula in the language of ZFC whose free variables are among x, y, A, w1, . . . , wn , so that in particular B is not free in ϕ. Then:

[ ( )] ∀A∀w1∀w2 ... ∀wn ∀x(x ∈ A ⇒ ∃!y ϕ) ⇒ ∃B ∀x x ∈ A ⇒ ∃y(y ∈ B ∧ ϕ) .

In other words, if the relation ϕrepresents a definable function f, A represents its domain, and f(x) is a set for every x in that domain, then the range of f is a subset of some set B . The form stated here, in which B may be larger than strictly necessary, is sometimes called the axiom schema of collection. 178 CHAPTER 44. ZERMELO–FRAENKEL SET THEORY

44.2.7 7. Axiom of infinity

Main article: Axiom of infinity

Let S(w)abbreviate w ∪ {w}, where w is some set (We can see that {w} is a valid set by applying the Axiom of Pairing with x = y = wso that the set zis {w}). Then there exists a set X such that the empty set ∅ is a member of X and, whenever a set y is a member of X, then S(y)is also a member of X.

∃X [∅ ∈ X ∧ ∀y(y ∈ X ⇒ S(y) ∈ X)] . More colloquially, there exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω, which can also be thought of as the set of natural numbers N .

44.2.8 8. Axiom of power set

Main article: Axiom of power set

By definition a set z is a subset of a set x if and only if every element of z is also an element of x:

(z ⊆ x) ⇔ (∀q(q ∈ z ⇒ q ∈ x)). The Axiom of Power Set states that for any set x, there is a set y that contains every subset of x:

∀x∃y∀z[z ⊆ x ⇒ z ∈ y]. The axiom schema of specification is then used to define the power set P(x) as the subset of such a y containing the subsets of x exactly:

P (x) = {z ∈ y : z ⊆ x} Axioms 1–8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed in Jech (2003). Some ZF axiomatizations include an axiom asserting that the empty set exists. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set x whose existence is being asserted are just those sets which the axiom asserts x must contain. The following axiom is added to turn ZF into ZFC:

44.2.9 9. Well-ordering theorem

Main article: Well-ordering theorem

For any set X, there is a binary relation R which well-orders X. This means R is a linear order on X such that every nonempty subset of X has a member which is minimal under R.

∀X∃R(R well-orders X). Given axioms 1–8, there are many statements provably equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let X be a set whose members are all non-empty. Then there exists a function f from X to the union of the members of X, called a "choice function", such that for all Y ∈ X one has f(Y) ∈ Y. Since the existence of a choice function when X is a finite set is easily proved from axioms 1–8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice set but says nothing about how the choice set is to be “constructed.” Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts. 44.3. MOTIVATION VIA THE CUMULATIVE HIERARCHY 179

44.3 Motivation via the cumulative hierarchy

One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann (Shoenfield 1977, sec. 2). In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0 there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2; see Hinman (2005, p. 467). The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V. It is provable that a set is in V if and only if the set is pure and well-founded; and provable that V satisfies all the axioms of ZFC, if the class of ordinals has appropriate reflection properties. For example, suppose that a set x is added at stage α, which means that every element of x was added at a stage earlier than α. Then every subset of x is also added at stage α, because all elements of any subset of x were also added before stage α. This means that any subset of x which the axiom of separation can construct is added at stage α, and that the powerset of x will be added at the next stage after α. For a complete argument that V satisfies ZFC see Shoenfield (1977). The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related ax- iomatic set theories such as Von Neumann–Bernays–Gödel set theory (often called NBG) and Morse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such as New Foundations. It is possible to change the definition of V so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more “narrow” hierarchy which gives the constructible universe L, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether V = L. Although the structure of L is more regular and well behaved than that of V, few mathematicians argue that V = L should be added to ZFC as an additional axiom.

44.4 Metamathematics

The axiom schemata of replacement and separation each contain infinitely many instances. Montague (1961) included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, Von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includes proper classes as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any theorem not mentioning classes and provable in one theory can be proved in the other. Gödel’s second incompleteness theorem says that a recursively axiomatizable system that can interpret Robinson arithmetic can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC. Hence the consistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of naive set theory: Russell’s paradox, the Burali-Forti paradox, and Cantor’s paradox. Abian and LaMacchia (1978) studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using models, they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because there are non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms. If consistent, ZFC cannot prove the existence of the inaccessible cardinals that category theory requires. Huge sets of this nature are possible if ZF is augmented with Tarski’s axiom (Tarski 1939). Assuming that axiom turns the axioms of infinity, power set, and choice (7 − 9 above) into theorems. 180 CHAPTER 44. ZERMELO–FRAENKEL SET THEORY

44.4.1 Independence

Many important statements are independent of ZFC (see list of statements undecidable in ZFC). The independence is usually proved by forcing, whereby it is shown that every countable transitive model of ZFC (sometimes augmented with large cardinal axioms) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms. Forcing proves that the following statements are independent of ZFC:

• Continuum hypothesis

• Diamond principle

• Suslin hypothesis

• Martin’s axiom (which is not a ZFC axiom)

• Axiom of Constructibility (V=L) (which is also not a ZFC axiom).

Remarks:

• The consistency of V=L is provable by inner models but not forcing: every model of ZF can be trimmed to become a model of ZFC + V=L.

• The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis.

• Martin’s axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.

• The constructible universe satisfies the Generalized Continuum Hypothesis, the Diamond Principle, Martin’s Axiom and the Kurepa Hypothesis.

• The failure of the Kurepa hypothesis is equiconsistent with the existence of a strongly inaccessible cardinal.

A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C. Another method of proving independence results, one owing nothing to forcing, is based on Gödel’s second in- completeness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel’s second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of large cardinals is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.

44.5 Criticisms

For criticism of set theory in general, see Objections to set theory

ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the universal set. Many mathematical theorems can be proven in much weaker systems than ZFC, such as Peano arithmetic and second order arithmetic (as explored by the program of reverse mathematics). Saunders Mac Lane and Solomon Feferman have both made this point. Some of “mainstream mathematics” (mathematics not directly connected with axiomatic 44.6. SEE ALSO 181 set theory) is beyond Peano arithmetic and second order arithmetic, but still, all such mathematics can be carried out in ZC (Zermelo set theory with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself. On the other hand, among axiomatic set theories, ZFC is comparatively weak. Unlike New Foundations, ZFC does not admit the existence of a universal set. Hence the universe of sets under ZFC is not closed under the elementary operations of the algebra of sets. Unlike von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory (MK), ZFC does not admit the existence of proper classes. These ontological restrictions are required for ZFC to avoid Russell’s paradox, but critics argue these restrictions make the ZFC axioms fail to capture the informal concept of set. A further comparative weakness of ZFC is that the axiom of choice included in ZFC is weaker than the included in MK. There are numerous mathematical statements undecidable in ZFC. These include the continuum hypothesis, the , and the Normal Moore space conjecture. Some of these conjectures are provable with the addition of axioms such as Martin’s axiom, large cardinal axioms to ZFC. Some others are decided in ZF+AD where AD is the axiom of determinacy, a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom (see projective determinacy). The Mizar system and Metamath have adopted Tarski–Grothendieck set theory, an extension of ZFC, so that proofs involving Grothendieck universes (encountered in category theory and algebraic geometry) can be formalized.

44.6 See also

• Foundation of mathematics

• Inner model

• Large cardinal axiom

Related axiomatic set theories:

• Morse–Kelley set theory

• Von Neumann–Bernays–Gödel set theory

• Tarski–Grothendieck set theory

• Constructive set theory

• Internal set theory

44.7 References

• Abian, Alexander (1965). The Theory of Sets and Transfinite Arithmetic. W B Saunders.

• ———; LaMacchia, Samuel (1978). “On the Consistency and Independence of Some Set-Theoretical Ax- ioms”. Notre Dame Journal of Formal Logic 19: 155–58. doi:10.1305/ndjfl/1093888220.

• Devlin, Keith (1996) [1984]. The Joy of Sets. Springer.

• Heinz-Dieter Ebbinghaus, 2007. Ernst Zermelo: An Approach to His Life and Work. Springer. ISBN 978-3- 540-49551-2.

• Abraham Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, 1973 (1958). Foundations of Set Theory. North- Holland. Fraenkel’s final word on ZF and ZFC.

• Hatcher, William, 1982 (1968). The Logical Foundations of Mathematics. Pergamon Press.

• Peter Hinman, 2005, Fundamentals of Mathematical Logic, A K Peters. ISBN 978-1-56881-262-5 182 CHAPTER 44. ZERMELO–FRAENKEL SET THEORY

• Thomas Jech, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2. • Kenneth Kunen, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

• Richard Montague, 1961, “Semantic closure and non-finite axiomatizability” in Infinistic Methods. London: Pergamon Press: 45–69.

• Patrick Suppes, 1972 (1960). Axiomatic Set Theory. Dover reprint. Perhaps the best exposition of ZFC before the independence of AC and the Continuum hypothesis, and the emergence of large cardinals. Includes many theorems. • Gaisi Takeuti and Zaring, W M, 1971. Introduction to Axiomatic Set Theory. Springer-Verlag.

• Alfred Tarski, 1939, “On well-ordered subsets of any set,”, Fundamenta Mathematicae 32: 176-83. • Tiles, Mary, 2004 (1989). The Philosophy of Set Theory. Dover reprint. Weak on metatheory; the author is not a mathematician.

• Tourlakis, George, 2003. Lectures in Logic and Set Theory, Vol. 2. Cambridge University Press. • Jean van Heijenoort, 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. Includes annotated English translations of the classic articles by Zermelo, Fraenkel, and Skolem bearing on ZFC.

• Zermelo, Ernst (1908). “Untersuchungen über die Grundlagen der Mengenlehre I”. Mathematische Annalen 65: 261–281. doi:10.1007/BF01449999. English translation in Heijenoort, Jean van (1967). “Investigations in the foundations of set theory”. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Source Books in the History of the Sciences. Harvard University Press. pp. 199–215. ISBN 978-0-674-32449-7.

• Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche”. Fundamenta Mathematicae 16: 29–47. ISSN 0016-2736.

44.8 External links

• Hazewinkel, Michiel, ed. (2001), “ZFC”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Stanford Encyclopedia of Philosophy articles by Thomas Jech:

• Set Theory; • Axioms of Zermelo–Fraenkel Set Theory.

• Metamath version of the ZFC axioms — A concise and nonredundant axiomatization. The background first order logic is defined especially to facilitate machine verification of proofs.

• A derivation in Metamath of a version of the separation schema from a version of the replacement schema. • Zermelo-Fraenkel Axioms at PlanetMath.org.

• Weisstein, Eric W., “Zermelo-Fraenkel Set Theory”, MathWorld. 44.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 183

44.9 Text and image sources, contributors, and licenses

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Night, LittleOldMe, Althai, LookingGlass, R'n'B, Ttwo, Policron, Szipucsu, Anony- mous Dissident, SieBot, J-puppy, Randomblue, ClueBot, DFRussia, MarceloTapiaGaete~enwiki, J567o678e789l, Thingg, XLinkBot, Чръный человек, Addbot, AnnaFrance, Tassedethe, Luckas-bot, Yobot, Fraggle81, TaBOT-zerem, TechKid, AnomieBOT, Rubinbot, Bsea, AdjustShift, Csigabi, Xqbot, Gonzalcg, Лев Дубовой, FrescoBot, TheLaeg, Kismalac, Distortiondude, RjwilmsiBot, Уральский Кот, Beyond My Ken, EmausBot, Set theorist, Tanner Swett, Josve05a, Quondum, Wmayner, L Kensington, ClueBot NG, Wcherowi, Demonsquirrel, Helpful Pixie Bot, Joydeep, Micenine, Khalpern, ChrisGualtieri, Jochen Burghardt, Mark viking, IORIUS, Weux082690, Nigellwh, Monkbot and Anonymous: 122 • Class (set theory) Source: http://en.wikipedia.org/wiki/Class_(set_theory)?oldid=656020610 Contributors: AxelBoldt, Tobias Hoevekamp, Dan~enwiki, The Anome, Taw, Toby Bartels, SimonP, Chinju, Ahoerstemeier, Andres, Schneelocke, Renamed user 4, Dcoetzee, Dys- prosia, Hyacinth, Bkell, Tobias Bergemann, Giftlite, Piotrus, Pmanderson, Sam Hocevar, Guanabot, Paul August, Zaslav, EmilJ, Randall Holmes, Mike Schwartz, .:Ajvol:., Red Winged Duck, Pion, Linas, Salix alba, R.e.b., Chobot, Roboto de Ajvol, YurikBot, Hairy Dude, Megapixie, GrinBot~enwiki, SmackBot, Rtc, Eskimbot, Pietaster, Acepectif, Mets501, CRGreathouse, CBM, Gregbard, Sam Staton, Egriffin, Bradediger, Dougher, JAnDbot, Frankie816, Jay Gatsby, Albmont, Ttwo, VolkovBot, Saibod, Broadbot, Tomaxer, Allebor- goBot, Keilana, Cenarium, Mlakc, Addbot, SpellingBot, MagnusA.Bot, Ginosbot, LinkFA-Bot, Lightbot, Luckas-bot, Yobot, , Xqbot, VladimirReshetnikov, ViolaPlayer, ComputScientist, Tkuvho, EmausBot, ZéroBot, BG19bot, AvocatoBot and Anonymous: 26 • Club filter Source: http://en.wikipedia.org/wiki/Club_filter?oldid=576827253 Contributors: WouterVH, Porcher, Salix alba, RussBot, Nate Biggs, SmackBot, Maksim-e~enwiki, Mets501, CBM, 777sms, Tonyxty and MerlIwBot • Club set Source: http://en.wikipedia.org/wiki/Club_set?oldid=607162243 Contributors: AxelBoldt, Rmhermen, Toby Bartels, WouterVH, Charles Matthews, Melikamp, Ben Standeven, Gauge, Carbon Caryatid, Linas, Porcher, Trovatore, SmackBot, Mets501, Zero sharp, JR- Spriggs, CBM, SwiftBot, Pavel Jelínek, YohanN7, Addbot, Yobot, MPeterHenry, 777sms, ZéroBot and Anonymous: 9 • Continuum hypothesis Source: http://en.wikipedia.org/wiki/Continuum_hypothesis?oldid=663811724 Contributors: AxelBoldt, Archibald Fitzchesterfield, Bryan Derksen, Zundark, The Anome, Andre Engels, Hari, Miguel~enwiki, Roadrunner, Shii, Stevertigo, Spiff~enwiki, Michael Hardy, Nixdorf, MartinHarper, Gabbe, Wapcaplet, Karada, Eric119, Snoyes, Marco Krohn, Tim Retout, Schneelocke, Ideyal, Charles Matthews, Timwi, Vanu, Reddi, Paul Stansifer, Doradus, .mau., Phil Boswell, Aleph4, Donarreiskoffer, Fredrik, Bethenco, Timrollpickering, Tobias Bergemann, Giftlite, Dbenbenn, Graeme Bartlett, Ævar Arnfjörð Bjarmason, Lethe, Fropuff, Dratman, Ee- quor, Fuzzy Logic, Icairns, Frenchwhale, Rich Farmbrough, TedPavlic, Guanabot, Luqui, Paul August, EmilJ, Robotje, Reinyday, Obradovic Goran, Haham hanuka, Crust, Jumbuck, Keenan Pepper, Sligocki, Adrian.benko, Oleg Alexandrov, Joriki, StradivariusTV, Ruud Koot, Isnow, Marudubshinki, Graham87, Jobnikon, Yurik, Porcher, Rjwilmsi, NatusRoma, Salix alba, R.e.b., Penumbra2000, FlaBot, Laubrau~enwiki, YurikBot, Open4D, Ksnortum, Gaius Cornelius, Abarry, Stassats, B-Con, CarlHewitt, SEWilcoBot, Trova- tore, Eltwarg, Arthur Rubin, AssistantX, GrinBot~enwiki, SmackBot, Nihonjoe, InverseHypercube, SaxTeacher, TimBentley, DHN- bot~enwiki, Tekhnofiend, Cybercobra, DRLB, Byelf2007, Loadmaster, Mets501, MedeaMelana, Quaeler, Dan Gluck, Zero sharp, JR- Spriggs, CBM, Discordant~enwiki, Gregbard, Cydebot, Peterdjones, Michael C Price, Thijs!bot, King Bee, Drpixie, Dugwiki, Eleuther, AntiVandalBot, M cuffa, Ling.Nut, Sullivan.t.j, David Eppstein, Kope, R'n'B, Alexcalamaro, Leocat, Ttwo, SpeedOfDarkness, VolkovBot, Red Act, Nxavar, Layman1, YohanN7, Dogah, SieBot, Ivan Štambuk, Alexis Humphreys, Likebox, Jojalozzo, Anchor Link Bot, CBM2, Cngoulimis, JustinBlank, JuPitEer, Hans Adler, Candhrim~enwiki, Jsondow, Thehelpfulone, MelonBot, Avoided, DOI bot, Unzerleg- barkeit, Lightbot, Know-edu, Legobot, Luckas-bot, Yobot, AnomieBOT, Materialscientist, Citation bot, ArthurBot, Xqbot, RJGray, VladimirReshetnikov, CES1596, Sémaphore, Citation bot 1, Tkuvho, Elockid, RedBot, Belovedeagle, 777sms, Pierpao, ClueBot NG, Deadwooddrz, Shivsagardharam, WhatisFGH, Trichometetrahydron, ChrisGualtieri, Ardegloo, Dexbot, Andyhowlett, Qualois, Adam.conkey, Monkbot, Wchargin, MathPhilFan, SoSivr and Anonymous: 113 • Critical point (set theory) Source: http://en.wikipedia.org/wiki/Critical_point_(set_theory)?oldid=491036359 Contributors: Zundark, Samw, Rich Farmbrough, Zero sharp, JRSpriggs, Ntsimp, Abtract, Hans Adler, Yobot, Erik9bot, Helpful Pixie Bot and Anonymous: 1 • Dana Scott Source: http://en.wikipedia.org/wiki/Dana_Scott?oldid=663496385 Contributors: Vkuncak, Tomo, Michael Hardy, Wap- caplet, Chinju, Markhurd, Hyacinth, Jaredwf, RedWolf, Tobias Bergemann, Giftlite, Markus Krötzsch, TonyW, Creidieki, Klemen Kocjancic, Asiananimal, D6, Leibniz, Bender235, Mkegelmann, BACbKA, Chalst, NotAbel, Nk, Zygmunt lozinski, Pearle, Snowolf, 184 CHAPTER 44. ZERMELO–FRAENKEL SET THEORY

Oleg Alexandrov, Etacar11, Kam Solusar, Ruud Koot, Bluemoose, Emerson7, Rjwilmsi, Lockley, Brighterorange, Corington, YurikBot, Koffieyahoo, CarlHewitt, Trovatore, Gareth Jones, RFBailey, Jpbowen, DYLAN LENNON~enwiki, SmackBot, Roger Hui, Sakhalinrf, Eskimbot, Slaniel, Alan smithee, Janm67, Tsca.bot, Nima Baghaei, Andrei Stroe, Ser Amantio di Nicolao, Syrcatbot, Genisock2, Doczilla, Jetman, Joey80, Pierre de Lyon, Cydebot, Thijs!bot, Faigl.ladislav, Wookiepedian, David Eppstein, Johnpacklambert, JamesD'Alexander, SparsityProblem, TXiKiBoT, Lampica, AlleborgoBot, Resurgent insurgent, JulesN, SieBot, Prof. Hewitt, Vojvodaen, DFRussia, Hairy- Fotr, Bender2k14, Kbdankbot, Addbot, LaaknorBot, Lightbot, Luckas-bot, Yobot, Lewix, Thore Husfeldt, Omnipaedista, Kiefer.Wolfowitz, TobeBot, RjwilmsiBot, EmausBot, Set theorist, Chimpionspeak, Suslindisambiguator, ClueBot NG, Helpful Pixie Bot, Adib5271, Liz, Dough34, Epi100, Jonarnold1985, KasparBot and Anonymous: 26 • Diagonal intersection Source: http://en.wikipedia.org/wiki/Diagonal_intersection?oldid=542564744 Contributors: Charles Matthews, ,.אנדריי ב ,Melikamp, Porcher, The Giant Puffin, Amakuru, JRSpriggs, David Eppstein, Matthew Yeager, Addbot, 777sms, ZéroBot Luizpuodzius and Makecat-bot • Elementary equivalence Source: http://en.wikipedia.org/wiki/Elementary_equivalence?oldid=655373914 Contributors: Michael Hardy, Charles Matthews, Waltpohl, Jabowery, Avn, Paul August, Bender235, Nortexoid, John Baez, Reetep, Algebraist, Wiki alf, Bbaumer, Mhss, Rsimmonds01, Physis, CRGreathouse, CBM, Marek69, Dricherby, DesolateReality, Hans Adler, MystBot, Addbot, Calle, Arthur- Bot, Drilnoth, Anne Bauval, Citation bot 1, Helpful Pixie Bot, Darvii and Anonymous: 5 • Equiconsistency Source: http://en.wikipedia.org/wiki/Equiconsistency?oldid=621680701 Contributors: Paul August, AndrewWTaylor, SmackBot, Bazonka, Aecea 1, JRSpriggs, CRGreathouse, CBM, Gregbard, Nick Number, Owlgorithm, PMLawrence, JRB-Europe, GamesMaxter, Omnipaedista, DrilBot, Xnn, RjwilmsiBot, Deltahedron and Anonymous: 7 • Extender (set theory) Source: http://en.wikipedia.org/wiki/Extender_(set_theory)?oldid=646723407 Contributors: Ben Standeven, Leyo, Hans Adler, Cnwilliams, Chimpionspeak, BG19bot and Mkoeberl • Fodor’s lemma Source: http://en.wikipedia.org/wiki/Fodor’{}s_lemma?oldid=637319395 Contributors: WouterVH, Charles Matthews, Tobias Bergemann, Giftlite, Ben Standeven, Oleg Alexandrov, Julien Tuerlinckx, Dpv, Porcher, FF2010, Zero sharp, Kope, Addbot, Yobot, GrouchoBot, VladimirReshetnikov, 777sms, EmausBot, ZéroBot, BG19bot, ArikVirus and Anonymous: 3 • Huge cardinal Source: http://en.wikipedia.org/wiki/Huge_cardinal?oldid=607163238 Contributors: The Anome, Toby Bartels, Takuya- Murata, Schneelocke, Dmytro, Giftlite, Loren36, Ben Standeven, Oleg Alexandrov, R.e.b., Trovatore, SmackBot, JRSpriggs, CBM, Headbomb, MetsBot, Kope, ClueBot, Jarble, Yobot, Citation bot, Brendanology, Andykim901 and Anonymous: 3 • Inaccessible cardinal Source: http://en.wikipedia.org/wiki/Inaccessible_cardinal?oldid=628976065 Contributors: Zundark, Toby Bar- tels, FvdP, Chinju, TakuyaMurata, Schneelocke, Timwi, Fibonacci, Tobias Bergemann, Lethe, Luqui, Ben Standeven, Shadow demon, Grick, Oleg Alexandrov, Cryo~enwiki, Eyu100, R.e.b., Chobot, Gaius Cornelius, Trovatore, Expensivehat, Crasshopper, Geoffrey.landis, SmackBot, Mets501, Stotr~enwiki, JRSpriggs, CBM, Jokes Free4Me, Michael C Price, Verkhovensky, David Eppstein, DeaconJohn- Fairfax, Tomas e, MystBot, Addbot, Citation bot, RJGray, Gonzalcg, TjBot, WikitanvirBot, ZéroBot, Mark viking, Blackbombchu and Anonymous: 13 • Ineffable cardinal Source: http://en.wikipedia.org/wiki/Ineffable_cardinal?oldid=543711107 Contributors: Vicki Rosenzweig, Takuya- Murata, Schneelocke, Charles Matthews, Aleph4, Tobias Bergemann, Eoghan, Ben Standeven, Oleg Alexandrov, Linas, R.e.b., GoOd- CoNtEnT, Mets501, JRSpriggs, Norman314, David Eppstein, LokiClock, Hans Adler, Addbot, Citation bot 1, ZéroBot and Anonymous: 5 • Intersection (set theory) Source: http://en.wikipedia.org/wiki/Intersection_(set_theory)?oldid=658548914 Contributors: AxelBoldt, Tarquin, Andre Engels, Toby~enwiki, Toby Bartels, Michael Hardy, Wshun, Booyabazooka, Den fjättrade ankan~enwiki, AugPi, Dpol, Andres, Hashar, Revolver, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Hyacinth, Donarreiskoffer, Robbot, Romanm, Bkell, Mil- losh, Tobias Bergemann, Giftlite, Fropuff, Sam Hocevar, Creidieki, Mormegil, Mindspillage, Rich Farmbrough, Hydrox, Dbachmann, Paul August, Elwikipedista~enwiki, EurekaLott, Haham hanuka, Jumbuck, Inky, Bookandcoffee, Kenyon, Mindmatrix, Apokrif, Is- now, Mandarax, Graham87, Salix alba, Chobot, YurikBot, ML, Ms2ger, Bodo Thiesen, Reedy, InverseHypercube, Melchoir, Eskimbot, Object01, Gilliam, Rmosler2100, Octahedron80, DHN-bot~enwiki, Antonrojo, Esproj, Cybercobra, Mets501, EdC~enwiki, Newone, Tophtucker, Devourer09, CBM, Mhaitham.shammaa, Pirtu, Spencer, Gökhan, JaGa, CommonsDelinker, Ttwo, VolkovBot, Anonymous Dissident, PaulTanenbaum, Synthebot, AlleborgoBot, SieBot, ToePeu.bot, Pyth~enwiki, Alexbot, Watchduck, Computer97, Bigoperm, Beroal, Addbot, Loupeter, Luckas-bot, Yobot, KamikazeBot, Hizkey, AnomieBOT, Clark89, Jsorr, TobeBot, DARTH SIDIOUS 2, Shafaet, EmausBot, ZéroBot, Chewings72, ChuispastonBot, ClueBot NG, Brad7777, Glacialfox, Lucifer bobby, Stephan Kulla, Brirush, YiFeiBot, Andrybak and Anonymous: 63 • Lebesgue measure Source: http://en.wikipedia.org/wiki/Lebesgue_measure?oldid=664087476 Contributors: AxelBoldt, Zundark, Miguel~enwiki, Patrick, Michael Hardy, TakuyaMurata, Loisel, Stevan White, Poor Yorick, Vargenau, Revolver, John Cross, Dcoetzee, Populus, Fi- bonacci, Tobias Bergemann, Pdenapo, Weialawaga~enwiki, Tosha, Giftlite, Lethe, MathKnight, Daniel Brockman, CSTAR, Vivacissama- mente, Rich Farmbrough, TedPavlic, Guanabot, Paul August, Bender235, Rbj, Obradovic Goran, Jumbuck, Uncle Bill, Oleg Alexandrov, Arneth, StradivariusTV, OdedSchramm, Graham87, JonathanZ, Chobot, Algebraist, YurikBot, Stormbay, Trovatore, CLW, Scriber~enwiki, Banus, SmackBot, MalafayaBot, Nbarth, Hongooi, Feraudyh, Naumz, Ejcaro, KazKylheku, CRGreathouse, CBM, David Cooke, E.G., Bernard the Varanid, PKT, King Bee, Vantelimus, Salgueiro~enwiki, Jazzam, JAnDbot, Alokbakshi, Sullivan.t.j, Americanhero, Kope, Maurice Carbonaro, Owlgorithm, O.mangold, AlleborgoBot, Paolo.dL, Nnemo, Grubb257, CàlculIntegral, Addbot, Lightbot, Legobot, Luckas-bot, Yobot, Lucubrations, AnomieBOT, Erel Segal, Bdmy, FrescoBot, Rckrone, Citation bot 1, DrilBot, Wham Bam Rock II, Chiqago, Maschen, CountMacula, ChuispastonBot, Raiden10, Rezabot, Jorgecarleitao, CitationCleanerBot, JYBot, Stephan Kulla, Frosty, Ronoth, SoSivr and Anonymous: 60 • Measurable cardinal Source: http://en.wikipedia.org/wiki/Measurable_cardinal?oldid=654910108 Contributors: Zundark, TakuyaMu- rata, Schneelocke, Charles Matthews, Dmytro, Chris Roy, MathMartin, Hadal, Tobias Bergemann, Giftlite, Fropuff, Perl, Ben Standeven, Gauge, Dfeldmann, Tabletop, OneWeirdDude, R.e.b., Trovatore, Crasshopper, Melanchthon, SmackBot, Aecea 1, Mets501, Stotr~enwiki, Zero sharp, JRSpriggs, CBM, DorganBot, VolkovBot, LokiClock, YohanN7, Hans Adler, Addbot, Citation bot, Gonzalcg, LucienBOT, Trappist the monk, DrBizarro, Chricho, Bomazi, Helpful Pixie Bot, Paolo Lipparini and Anonymous: 10 • Measure (mathematics) Source: http://en.wikipedia.org/wiki/Measure_(mathematics)?oldid=665931809 Contributors: AxelBoldt, Zun- dark, Iwnbap, Andre Engels, Toby~enwiki, Toby Bartels, Miguel~enwiki, Patrick, Michael Hardy, Gabbe, TakuyaMurata, Loisel, Loren Rosen, Revolver, Charles Matthews, Dino, Dysprosia, Prumpf, Fibonacci, Robbot, Gandalf61, MathMartin, Sverdrup, Aetheling, Ru- akh, Tobias Bergemann, Pdenapo, Weialawaga~enwiki, Ancheta Wis, Tosha, Giftlite, Mousomer, BenFrantzDale, Lethe, Lupin, Math- Knight, Everyking, Mike40033, Uranographer, OverlordQ, CSTAR, Pmanderson, Vivacissamamente, PhotoBox, Keenanpepper, Dis- cospinster, Rich Farmbrough, Gadykozma, Mat cross, Harriv, Paul August, Bender235, Elwikipedista~enwiki, Gauge, Rgdboer, Jung 44.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 185

dalglish, Gar37bic, 3mta3, Obradovic Goran, Tsirel, Msh210, Alansohn, ABCD, Caesura, Jheald, AiusEpsi, Oleg Alexandrov, Feezo, Joriki, DealPete, Linas, Isnow, BD2412, Dpv, MarSch, Salix alba, Mike Segal, R.e.b., FlaBot, Mathbot, Chobot, DVdm, Bgwhite, Roboto de Ajvol, YurikBot, RMcGuigan, Manop, Gaius Cornelius, Rat144, Crasshopper, Bota47, Rktect, Googl, Brian Tvedt, Mebden, Benandorsqueaks, GrinBot~enwiki, Zvika, Finell, SmackBot, Eskimbot, Gilliam, SchfiftyThree, MrRage, RayAYang, Nbarth, DHN- bot~enwiki, Geevee, Foxjwill, Turms, EIFY, Henning Makholm, Jóna Þórunn, Lambiam, Richard L. Peterson, The Infidel, Irvin83, 16@r, Ashigabou, Levineps, Zero sharp, Markjoseph125, CRGreathouse, Jackzhp, CBM, Matthew Auger, Thomasmeeks, Myasuda, Xantharius, Omicronpersei8, Thijs!bot, Edokter, Salgueiro~enwiki, JAnDbot, MER-C, Takwan, Avaya1, Arvinder.virk, Beaumont, Ro- gierBrussee, Jay Gatsby, Albmont, GIrving, Sullivan.t.j, David Eppstein, Akulo, Cdamama, Daniele.tampieri, Policron, Juliancolton, VolkovBot, Hesam7, Saibod, Digby Tantrum, PaulTanenbaum, Geometry guy, Piyush Sriva, Ocsenave, SieBot, Stca74, Boobahmad101, BrianS36, Paolo.dL, Mimihitam, Thehotelambush, Jorgen W, Baaaaaaar, S2000magician, Melcombe, Ken123BOT, The Thing That Should Not Be, MABadger, Lbertolotti, Masterpiece2000, Sun Creator, 7&6=thirteen, Nicoguaro, SilvonenBot, Addbot, AndersBot, Yobot, AnomieBOT, Bdmy, Dowjgyta, Ptrf, Jsharpminor, Omnipaedista, Point-set topologist, Charvest, Semistablesystem, FrescoBot, Dave Ordinary, Sławomir Biały, Citation bot 1, Zhangkai Jason Cheng, Dark Charles, RandomDSdevel, Kiefer.Wolfowitz, Yahia.barie, Danielbojczuk, Pokus9999, Tcnuk, Bgpaulus, Le Docteur, AleHitch, Boplin, Slawekb, Empty Buffer, ClueBot NG, Michael P. Barnett, Frietjes, Joel B. Lewis, Finanzmaster, MerlIwBot, Helpful Pixie Bot, KLBot2, Boriaj, HilberTraum, Brad7777, Randomguess, Chris- Gualtieri, Avastration, Acehole60, Stephan Kulla, Limit-theorem, Mark viking, Tentinator, Clebor42, K9re11, Suelru, KasparBot and Anonymous: 129 • Mitchell order Source: http://en.wikipedia.org/wiki/Mitchell_order?oldid=608656512 Contributors: Ben Standeven, Rjwilmsi, JRSpriggs, Kope, Yobot, Nima1024 and BattyBot • Model theory Source: http://en.wikipedia.org/wiki/Model_theory?oldid=664684414 Contributors: Damian Yerrick, Zundark, The Anome, Iwnbap, Toby Bartels, Youandme, Michael Hardy, David Martland, MartinHarper, Bcrowell, Chinju, Eric119, Snoyes, Rotem Dan, Re- volver, Charles Matthews, Dcoetzee, Dysprosia, Hyacinth, Fred ulisses maranhão, Robbot, Josh Cherry, Sparky, MathMartin, Henrygb, Halibutt, Tobias Bergemann, Filemon, Ancheta Wis, Giftlite, Lupin, Karl-Henner, Sam Hocevar, Creidieki, Smimram, Rcog, Paul Au- gust, Elwikipedista~enwiki, Gauge, Floorsheim, Chalst, EmilJ, Peter M Gerdes, Jonsafari, Mdd, Msh210, EmmetCaulfield, Trylks, Jim O'Donnell, Oleg Alexandrov, Joriki, Linas, Ruud Koot, Payrard, RuM, Graham87, BD2412, Grammarbot, R.e.b., JonathanZ, SDaniel, Tillmo, Algebraist, YurikBot, Hairy Dude, NTBot~enwiki, Archelon, CarlHewitt, NawlinWiki, Trovatore, Bota47, Ott2, Arthur Rubin, Sardanaphalus, SmackBot, Rtc, Melchoir, SaxTeacher, Ppntori, Spellchecker, Charles Moss, Viebel, Tompsci, Jon Awbrey, Byelf2007, Lambiam, Loadmaster, WAREL, Jason.grossman, Lottamiata, Zero sharp, Cyrusc, James pic, CRGreathouse, CBM, Gregbard, Gryakj, Blaisorblade, Julian Mendez, Bernard the Varanid, Klausness, Escarbot, WinBot, Majorly, Avaya1, David Eppstein, Thehalfone, Joejun- sun, Pavel Jelínek, Joshua Davis, Pomte, Tomaz.slivnik, Policron, Dessources, Thefrettinghand, VolkovBot, JohnBlackburne, Danadocus, Popopp, Synthebot, Sapphic, Logan, Barkeep, SieBot, Shellgirl, Pseudonomous, Mrw7, DesolateReality, Valeria.depaiva, WikiSBTR, Pi zero, Mild Bill Hiccup, Jusdafax, Bender2k14, Hans Adler, Qwfp, Little Mountain 5, Addbot, MagnusA.Bot, Loupeter, Wireless friend, Luckas-bot, Yobot, Ht686rg90, Pcap, AnomieBOT, Materialscientist, Citation bot, ArthurBot, Groovenstein, VladimirReshet- nikov, SassoBot, Ringspectrum, FrescoBot, LucienBOT, Paine Ellsworth, Citation bot 1, Tkuvho, Foobarnix, Gf uip, John of Reading, Stephan Spahn, Masssly, Ultracoffee, Helpful Pixie Bot, Itzuvit, Freitagj, Brad7777, Thierry Le Provost, Monkbot, SolidPhase, SoSivr, SocraticOath and Anonymous: 96 • Normal measure Source: http://en.wikipedia.org/wiki/Normal_measure?oldid=607169374 Contributors: Charles Matthews, JRSpriggs, CBM, Hans Adler, Yobot and Erik9bot • Ordinal number Source: http://en.wikipedia.org/wiki/Ordinal_number?oldid=666064832 Contributors: AxelBoldt, Mav, Bryan Derk- sen, Zundark, The Anome, Iwnbap, LA2, Christian List, B4hand, Olivier, Stevertigo, Patrick, Michael Hardy, Llywrch, Jketola, Chinju, TakuyaMurata, Karada, Docu, Vargenau, Revolver, Charles Matthews, Dysprosia, Malcohol, Owen, Rogper~enwiki, Hmackiernan, Bald- hur, Adhemar, Fuelbottle, Tobias Bergemann, Giftlite, Markus Krötzsch, Ævar Arnfjörð Bjarmason, Lethe, Fropuff, Gro-Tsen, That- tommyhall, Jorend, Siroxo, Wmahan, Beland, Joeblakesley, Elroch, 4pq1injbok, Luqui, Silence, Paul August, EmilJ, Babomb, Randall Holmes, Wood Thrush, Robotje, Blotwell, Crust, Jumbuck, Sligocki, SidP, DV8 2XL, Jim Slim, Oleg Alexandrov, Warbola, Linas, Miaow Miaow, Graham87, Grammarbot, Jorunn, Rjwilmsi, Bremen, Salix alba, Mike Segal, R.e.b., FlaBot, Jak123, Chobot, YurikBot, Wave- length, RobotE, Hairy Dude, CanadianCaesar, Archelon, Gaius Cornelius, Trovatore, Crasshopper, DeadEyeArrow, Pooryorick~enwiki, Hirak 99, Closedmouth, Arthur Rubin, PhS, GrinBot~enwiki, Brentt, Nicholas Jackson, SmackBot, Pokipsy76, KocjoBot~enwiki, Blue- bot, AlephNull~enwiki, Jiddisch~enwiki, Dreadstar, Mmehdi.g, Lambiam, Khazar, Minna Sora no Shita, Bjankuloski06en~enwiki, 16@r, Loadmaster, Limaner, Quaeler, Jason.grossman, Joseph Solis in Australia, Easwaran, Zero sharp, Tawkerbot2, JRSpriggs, Vaughan Pratt, CRGreathouse, CBM, Gregbard, FilipeS, HdZ, Pcu123456789, Lyondif02, Odoncaoa, Jj137, Hannes Eder, Shlomi Hillel, JAnDbot, Agol, BrentG, Smartcat, Bongwarrior, Swpb, David Eppstein, Jondaman21, R'n'B, IPonomarev, RockMFR, Ttwo, It Is Me Here, Policron, VolkovBot, Dommedagsprofet, Hotfeba, Jeff G., LokiClock, PMajer, Alphaios~enwiki, Cremepuff222, Wikithesource, Arcfrk, SieBot, Mrw7, J-puppy, TheCatalyst31, ClueBot, DFRussia, DanielDeibler, DragonBot, Hans Adler, StevenDH, Lacce, Against the current, Dthomsen8, Addbot, Dyaa, Mathemens, Unzerlegbarkeit, Luckas-bot, Yobot, Utvik old, THEN WHO WAS PHONE?, KamikazeBot, AnomieBOT, Angry bee, Citation bot, Nexx892, Twri, Xqbot, Freebirth Toad, Capricorn42, RJGray, GrouchoBot, VladimirReshetnikov, SassoBot, Citation bot 1, RedBot, Burritoburritoburrito, Thestraycat57, Raiden09, EmausBot, Fly by Night, Jens Blanck, SporkBot, Clue- Bot NG, Frietjes, Rezabot, Helpful Pixie Bot, BG19bot, Anthony.de.almeida.lopes, Jochen Burghardt, Mark viking, Pop-up casket, Jose Brox, The Horn Blower, George8211, Dconman2, Garfield Garfield, Lalaloopsy1234, SoSivr, Eth450, Neposner, Mircea BRT, Smwrd and Anonymous: 139 • Partially ordered set Source: http://en.wikipedia.org/wiki/Partially_ordered_set?oldid=663537797 Contributors: Bryan Derksen, Zun- dark, Tomo, Patrick, Bcrowell, Chinju, TakuyaMurata, GTBacchus, AugPi, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Doradus, Maximus Rex, Fibonacci, Tobias Bergemann, Giftlite, Markus Krötzsch, Fropuff, Peruvianllama, Jason Quinn, Neilc, Gubbubu, De- fLog~enwiki, MarkSweep, Urhixidur, TheJames, Paul August, Zaslav, Spoon!, Porton, Haham hanuka, DougOrleans, Msh210, Oleg Alexandrov, Daira Hopwood, MFH, Salix alba, FlaBot, Vonkje, Chobot, Laurentius, Dmharvey, Vecter, Sanguinity, Modify, RDBury, Incnis Mrsi, Brick Thrower, Cesine, Zanetu, Jcarroll, Nbarth, Jdthood, Javalenok, Kjetil1001, Dreadstar, Esoth~enwiki, Mike Fikes, A. Pichler, Vaughan Pratt, CRGreathouse, L'œuf, CBM, Werratal, Rlupsa, CZeke, Ill logic, JAnDbot, MER-C, BrotherE, Tbleher, A3nm, David Eppstein, SlamDiego, Bissinger, Haseldon, Daniel5Ko, GaborLajos, NewEnglandYankee, Orphic, RobertDanielEmerson, TXiK- iBoT, Digby Tantrum, PaulTanenbaum, Arcfrk, SieBot, Mochan Shrestha, TheGhostOfAdrianMineha, Thehotelambush, Megaloxantha, Peiresc~enwiki, Cheesefondue, Jludwig, ClueBot, Morinus, Justin W Smith, Methossant, Pi zero, Jonathanrcoxhead, Watchduck, Com- puterGeezer, He7d3r, Hans Adler, Jtle515, Palnot, Marc van Leeuwen, Ankan babee, Addbot, Download, Luckyz, Legobot, Kilom691, AnomieBOT, Erel Segal, Citation bot, SteveWoolf, Undsoweiter, FrescoBot, Nicolas Perrault III, Confluente, Ricardo Ferreira de Oliveira, Throw it in the Fire, Gnathan87, Setitup, EmausBot, John of Reading, Febuiles, Thecheesykid, ZéroBot, Chharvey, The man who was 186 CHAPTER 44. ZERMELO–FRAENKEL SET THEORY

Friday, SporkBot, Zfeinst, Rathgemz, CocuBot, Vdamanafshan, Mesoderm, MerlIwBot, Wbm1058, Jakshap, Paolo Lipparini, ElphiBot, Larion Garaczi, Aabhis, Jochen Burghardt, Mark viking, Eamonford, Sgbmyr, K401sTL3, Tudor987, Victor Lesyk, Some1Redirects4You and Anonymous: 77 • Power set Source: http://en.wikipedia.org/wiki/Power_set?oldid=655854839 Contributors: AxelBoldt, Zundark, Tarquin, Awaterl, Boleslav Bobcik, Michael Hardy, Wshun, TakuyaMurata, GTBacchus, Pcb21, Mxn, Charles Matthews, Berteun, Dysprosia, Jay, Hyacinth, Ed g2s, .mau., Aleph4, Robbot, Tobias Bergemann, Adam78, Giftlite, Dratman, DefLog~enwiki, Tomruen, Mormegil, Rich Farmbrough, Ted- Pavlic, Paul August, Zaslav, Elwikipedista~enwiki, Spayrard, SgtThroat, Obradovic Goran, Jumbuck, Kocio, Tony Sidaway, Ultramarine, Kenyon, Oleg Alexandrov, Linas, Flamingspinach, GregorB, Yurik, Salix alba, FlaBot, VKokielov, SchuminWeb, Small potato, Cia- Pan, NevilleDNZ, Chobot, YurikBot, Stephenb, Trovatore, Bota47, Deville, Closedmouth, MathsIsFun, Realkyhick, GrinBot~enwiki, SmackBot, InverseHypercube, Persian Poet Gal, SMP, Alink, Octahedron80, Kostmo, Armend, Shdwfeather, 16@r, Mike Fikes, Malter, Freelance Intellectual, JRSpriggs, Vaughan Pratt, CBM, Gregbard, Sam Staton, Goldencako, DumbBOT, Cj67, Abu-Fool Danyal ibn Amir al-Makhiri, Felix C. Stegerman, David Eppstein, R'n'B, RJASE1, UnicornTapestry, VolkovBot, Camrn86, Anonymous Dissident, PaulTanenbaum, Dmcq, AlleborgoBot, Pcruce, Faradayplank, MiNombreDeGuerra, Megaloxantha, KrustallosIce28, S2000magician, Classicalecon, Dmitry Dzhus, PipepBot, DragonBot, He7d3r, Marc van Leeuwen, Addbot, Freakmighty, Download, Luckas-bot, Yobot, Ht686rg90, ArthurBot, La Mejor Ratonera, FrescoBot, Showgun45, ComputScientist, Throw it in the Fire, Tkuvho, HRoestBot, El- Lutzo, John of Reading, WikitanvirBot, Lunaibis, Set theorist, Josve05a, AMenteLibera, Wcherowi, Helpful Pixie Bot, Sebastien.noir, Deltahedron, QuantumNico, Mark viking, Jadiker, Rajiv1965 and Anonymous: 87 • Preorder Source: http://en.wikipedia.org/wiki/Preorder?oldid=611521876 Contributors: AxelBoldt, Patrick, Repton, Delirium, An- dres, Dysprosia, Greenrd, Big Bob the Finder, BenRG, Tobias Bergemann, Giftlite, Markus Krötzsch, Lethe, Fropuff, Vadmium, De- fLog~enwiki, Zzo38, Jh51681, Barnaby dawson, Paul August, EmilJ, Msh210, Melaen, Joriki, Linas, Dionyziz, Mandarax, Salix alba, Cjoev, VKokielov, Mathbot, Jrtayloriv, YurikBot, Laurentius, Hairy Dude, WikidSmaht, Trovatore, Modify, Netrapt, Wasseralm, Smack- Bot, XudongGuan~enwiki, DCary, Jdthood, Mets501, PaulGS, Stotr~enwiki, Zero sharp, CRGreathouse, Michael A. White, David Eppstein, Jwuthe2, PaulTanenbaum, SieBot, Thehotelambush, Functor salad, He7d3r, Sun Creator, Cenarium, 1ForTheMoney, Palnot, Мыша, Legobot, Luckas-bot, AnomieBOT, DannyAsher, Xqbot, VladimirReshetnikov, ComputScientist, BrideOfKripkenstein, Noted- grant, WikitanvirBot, Lclem, Dfabera, SporkBot, RichardMills65, Khazar2, Lerutit, Jochen Burghardt, Reatlas, Damonamc and Anony- mous: 26 • Ramsey cardinal Source: http://en.wikipedia.org/wiki/Ramsey_cardinal?oldid=644349375 Contributors: Schneelocke, Charles Matthews, Dmytro, Aleph4, Tobias Bergemann, Ben Standeven, EmilJ, R.e.b., Vclaw, Trovatore, SmackBot, JRSpriggs, Myasuda, S Marshall, Aer- vanath, SieBot, Hans Adler, Addbot, Citation bot, Anne Bauval, Trappist the monk, ZéroBot, Mirror symmetry and Anonymous: 1 • Rank-into-rank Source: http://en.wikipedia.org/wiki/Rank-into-rank?oldid=583692940 Contributors: Zundark, Michael Hardy, Takuya- Murata, Julesd, Schneelocke, Charles Matthews, Jitse Niesen, Dmytro, Gene Ward Smith, Oleg Alexandrov, Josh Parris, Rjwilmsi, R.e.b., Trovatore, Luminus1, JRSpriggs, CBM, Headbomb, Widefox, Hans Adler, AnomieBOT, Citation bot, Citation bot 1 and Anonymous: 5 • Regular cardinal Source: http://en.wikipedia.org/wiki/Regular_cardinal?oldid=647231227 Contributors: Patrick, Dcoetzee, Dysprosia, Fibonacci, Choni, Tobias Bergemann, Lethe, Oleg Alexandrov, Rjwilmsi, OneWeirdDude, JonathanZ, Chobot, Trovatore, SmackBot, Tsca.bot, Turms, JRSpriggs, JAnDbot, Kope, Hurkyl, Austinmohr, DeaconJohnFairfax, The Thing That Should Not Be, Addbot, Unzer- legbarkeit, AnomieBOT, Gongfarmerzed, EmausBot, ZéroBot, Helpful Pixie Bot, Mark viking, François Robere, Lgidwani and Anony- mous: 9 • Scott’s trick Source: http://en.wikipedia.org/wiki/Scott’{}s_trick?oldid=657667164 Contributors: Michael Hardy, Tobias Bergemann, Giftlite, R.e.b., Algebraist, CBM, Unzerlegbarkeit, Yobot, Paolo Lipparini and Anonymous: 1 • Sequence Source: http://en.wikipedia.org/wiki/Sequence?oldid=665140592 Contributors: AxelBoldt, Mav, Zundark, Tarquin, XJaM, Toby Bartels, Imran, Camembert, Youandme, Lir, Patrick, Michael Hardy, Ihcoyc, Poor Yorick, Nikai, EdH, Charles Matthews, Dys- prosia, Greenrd, Hyacinth, Zero0000, Sabbut, Garo, Robbot, Lowellian, MathMartin, Stewartadcock, Henrygb, Bkell, Tosha, Centrx, Giftlite, BenFrantzDale, Lupin, Herbee, Horatio, Edcolins, Vadmium, Leonard Vertighel, Manuel Anastácio, Alexf, Fudo, Melikamp, Sam Hocevar, Tsemii, Ross bencina, Jiy, TedPavlic, Paul August, JoeSmack, Elwikipedista~enwiki, Syp, Pjrich, Shanes, Jonathan Drain, Nk, Obradovic Goran, Haham hanuka, Zaraki~enwiki, Merope, Jumbuck, Reubot, Jet57, Olegalexandrov, Ringbang, Djsasso, Total- cynic, Oleg Alexandrov, Hoziron, Linas, Madmardigan53, MFH, Isnow, Graham87, Dpv, Mendaliv, Salix alba, Figs, VKokielov, Log- gie, Rsenington, RexNL, Pexatus, Fresheneesz, Kri, Ryvr, Chobot, Lightsup55, Krishnavedala, Wavelength, Michael Slone, Grubber, Arthur Rubin, JahJah, Pred, Finell, KHenriksson, Gelingvistoj, Chris the speller, Bluebot, Nbarth, Mcaruso, Suicidalhamster, SundarBot, Dreadstar, Fagstein, Just plain Bill, Xionbox, Dreftymac, Gco, CRGreathouse, CBM, Gregbard, Cydebot, Xantharius, Epbr123, KCliffer, Saber Cherry, Rlupsa, Marek69, Urdutext, Icep, Ste4k, Mutt Lunker, JAnDbot, Asnac, Coolhandscot, Martinkunev, VoABot II, Avjoska, JamesBWatson, Brusegadi, Minimiscience, Stdazi, DerHexer, J.delanoy, Trusilver, Suenm~enwiki, Ncmvocalist, Belovedfreak, Policron, JingaJenga, VolkovBot, ABF, AlnoktaBOT, Philip Trueman, Digby Tantrum, JhsBot, Isis4563, Wolfrock, Xiong Yingfei, Newbyguesses, SieBot, Scarian, Yintan, Xelgen, Outs, Paolo.dL, OKBot, Pagen HD, Wahrmund, Classicalecon, Atif.t2, Crambo0349, ClueBot, Justin W Smith, Fyyer, SuperHamster, Excirial, Estirabot, Jotterbot, Thingg, Downgrader, Aj00200, XLinkBot, Stickee, Rror, WikHead, Brent- smith101, Addbot, Non-dropframe, Kongr43gpen, Matěj Grabovský, Legobot, Luckas-bot, Yobot, Eric-Wester, 4th-otaku, AnomieBOT, Jim1138, Law, Materialscientist, E2eamon, ArthurBot, Ayda D, Xqbot, Omnipaedista, RibotBOT, Charvest, Shadowjams, Thehelpful- bot, Dan6hell66, Constructive editor, Mark Renier, Tal physdancer, SixPurpleFish, Pinethicket, BRUTE, SkyMachine, PiRSquared17, Roy McCoy, RjwilmsiBot, Tzfyr, EmausBot, John of Reading, GoingBatty, Wikipelli, K6ka, Brent Perreault, Nellandmice, Bethnim, Ida Shaw, Alpha Quadrant, KuduIO, D.Lazard, SporkBot, Wayne Slam, Donner60, Chewings72, ClueBot NG, Satellizer, Widr, MerlI- wBot, Helpful Pixie Bot, HMSSolent, Curb Chain, Calabe1992, Brad7777, Minsbot, Praxiphenes, EuroCarGT, Ven Seyranyan., Jegyao, DavyRalph, Graphium, Jochen Burghardt, Brirush, Mark viking, LoMaPh, Immonster, EricsonWillians, Emlynlee, Buscus 3, JackHoang, Some1Redirects4You and Anonymous: 209 • Set theory Source: http://en.wikipedia.org/wiki/Set_theory?oldid=665506239 Contributors: AxelBoldt, Bryan Derksen, Zundark, The Anome, Christian List, Toby Bartels, Enchanter, Michael Hardy, Karada, William M. Connolley, Plaudite~enwiki, Andres, Evercat, Mar- cosantonio, Dysprosia, ThomasStrohmann~enwiki, HappyDog, Hyacinth, VeryVerily, Robbot, Fredrik, Peak, Romanm, Tobias Berge- mann, Giftlite, Gene Ward Smith, Bogdanb, Mintleaf~enwiki, Lethe, Siroxo, Python eggs, Andycjp, Crawdaddio, Sam Hocevar, Marcos, Mubor, David Sneek, EugeneZelenko, Discospinster, Zaheen, Vsmith, Mani1, Harriv, Paul August, Bender235, *drew, Rgdboer, Randall Holmes, Mike Schwartz, Giraffedata, Obradovic Goran, Msh210, Tablizer, Pinar, Max Naylor, Dirac1933, Itsmine, Feezo, Slgrand- son, Chun-hian, Mendaliv, Search4Lancer, Rjwilmsi, Koavf, Salix alba, R.e.b., JonnyR, Gurch, BMF81, Chobot, Pip2andahalf, RussBot, CarlHewitt, Rick Norwood, Joth, Dysmorodrepanis~enwiki, Trovatore, PrologFan, Jpbowen, Crasshopper, Wheelybrook, Haemo, Googl, 44.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 187

Arthur Rubin, Modify, Sardanaphalus, SmackBot, Unschool, Jagged 85, Brick Thrower, Skizzik, Dugas, Jprg1966, MalafayaBot, Darth Panda, MyNameIsVlad, Nick Levine, Glloq, KevM, Lesnail, Grover cleveland, Jiddisch~enwiki, Chrylis, Wvbailey, JorisvS, Rundquist, Bjankuloski06en~enwiki, Jim.belk, Dfass, Mr. Vernon, Loadmaster, Mets501, Iridescent, Quantum Burrito, Clarityfiend, Tophtucker, RekishiEJ, Supertigerman, Mjohnrussell, Spdegabrielle, CRGreathouse, CBM, MarsRover, Gregbard, Dragonflare82, Thijs!bot, J. W. Love, RichardVeryard, Klausness, Urdutext, Escarbot, AntiVandalBot, Emeraldcityserendipity, LibLord, Salgueiro~enwiki, Perelaar, Knotwork, JAnDbot, MER-C, Bongwarrior, Soulbot, Animum, David Eppstein, Kope, Khalid Mahmood, Musictheorist, Pharaoh of the Wizards, Maurice Carbonaro, NewEnglandYankee, Policron, Cometstyles, Tellerman, DorganBot, Treisijs, Idioma-bot, Alan U. Ken- nington, ABF, JohnBlackburne, LokiClock, AlnoktaBOT, TXiKiBoT, Red Act, Rei-bot, Anonymous Dissident, Saibod, Digby Tantrum, Kmitchell19, Synthebot, VanishedUserABC, AlleborgoBot, Jtarr, YohanN7, Dogah, SieBot, BotMultichill, Gerakibot, Viskonsas, Dis- neyfreek, Mnlkpo, MacPerry, Billyg, Yoda of Borg, CBM2, Neurophysics, B J Bradford, ClueBot, Smithpith, Cliff, DragonBot, Bear- Machine, Cenarium, Njardarlogar, 7&6=thirteen, Hans Adler, Palnot, BodhisattvaBot, JinJian, Multipundit, Deineka, Addbot, Manuel Trujillo Berges, Blake de doosten, Betterusername, Renamed user 5, Topology Expert, Ronhjones, NjardarBot, MrOllie, Download, Dyaa, Ozob, Bob K31416, Numbo3-bot, Tide rolls, TeH nOmInAtOr, Rrmsjp, DaveChild, Ben Ben, Legobot, Tartarus, Yobot, Frag- gle81, TaBOT-zerem, Pcap, Synchronism, Galoubet, Faizulhasan, OllieFury, Tbvdm, Xqbot, RJGray, Qualitydemise, Solphusion~enwiki, Point-set topologist, WissensDürster, La Mejor Ratonera, IShadowed, Red van man, Joxemai, Bekus, FrescoBot, Tobby72, Mark Re- nier, Dagme, Citation bot 1, FirmBenevolence, Tkuvho, Boulaur, Jschnur, Foobarnix, MPeterHenry, JumpDiscont, DARTH SIDIOUS 2, EmausBot, Set theorist, Jmencisom, Kimartz, Wikipelli, Midas02, Wayne Slam, Computationalverb, ChuispastonBot, JonRichfield, ClueBot NG, Wcherowi, SusikMkr, Movses-bot, The Master of Mayhem, MerlIwBot, Ftonti, Helpful Pixie Bot, MKar, Hallows AG, Xosé Antonio, Ingmar.lippert, Knwlgc, Nisse Phelsum, YatharthROCK, Brad7777, Sofia karampataki, Spasoev, Justincheng12345-bot, Alfasst, Deltahedron, Nandanchoudhary05, Mark viking, Flimflam97, William2001, Howicus, Tentinator, NeapleBerlina, SakeUPenn, K401sTL3, Mario Castelán Castro, Galileo9, Sashasct, Fahim Of Wiki and Anonymous: 209 • Stanislaw Ulam Source: http://en.wikipedia.org/wiki/Stanislaw_Ulam?oldid=666057135 Contributors: Kpjas, WojPob, Css, Andre En- gels, Vignaux, XJaM, Enchanter, Ray Van De Walker, Patrick, Michael Hardy, Rp, Gabbe, Chinju, IZAK, Paul A, Asparagirl, Charles Matthews, Dino, El~enwiki, Zoicon5, Maximus Rex, Bartosz, Topbanana, Aleph4, Jaredwf, Altenmann, Halibutt, Bkell, Cautious, Jer- ryFriedman, Ancheta Wis, Giftlite, Yeti~enwiki, MaGioZal, Fastfission, Pashute, Andrea Parri, Piotrus, Emax, Balcer, Pmanderson, Fin- tor, JamesTeterenko, ELApro, Dr.frog, D6, Jayjg, Rdb, Rich Farmbrough, Guanabot, Bender235, Circeus, Ruszewski, John Vandenberg, C S, MPerel, Nsaa, Lysdexia, Logologist, Burn, Bbsrock, VivaEmilyDavies, Oleg Alexandrov, Ashujo, Chinmin~enwiki, Kbdank71, Rjwilmsi, Koavf, Valentinejoesmith, Lockley, Jivecat, Amire80, Vegaswikian, R.e.b., Bubba73, FlaBot, Mathbot, Nihiltres, Witkacy, Brettbergeron, Tedder, Chobot, Volunteer Marek, SujinYH, YurikBot, Spacepotato, Encyclops, StuffOfInterest, RussBot, Hawkeye7, Wiki alf, Ospalh, Syd Henderson, Mareklug, Avraham, Curpsbot-unicodify, SmackBot, Roger Hui, Incnis Mrsi, Prodego, Jacek Kendysz, KocjoBot~enwiki, Stepa, Hmains, Sbharris, Mhym, Threeafterthree, Huon, Daqu, Jon Awbrey, Moulton, Thomaspaine, John, Mathiasrex, Vumba, Alpha Omicron, Norm mit, OS2Warp, CRGreathouse, Kowalmistrz~enwiki, Ken Gallager, Myasuda, Cydebot, Zee zack, Stud- erby, Master son, Eubulide, Thijs!bot, Epbr123, Kubanczyk, Konradek, Mibelz, Massimo Macconi, MECU, Husond, Avaya1, Matthew Fennell, Beaumont, .anacondabot, T@nn, Kosmopolis, Waacstats, Ali'i, Torchiest, David Eppstein, LorenzoB, Grantsky, Kskowron, R'n'B, WFinch, Aboutmovies, Haseldon, Plindenbaum, Idioma-bot, Kirkaiya, RaulCovita, Philip Trueman, Theophilus reed, Wasser- mann~enwiki, Gilisa, M0RD00R, Kacser, Dictioneer, AlleborgoBot, Semifinalist, PeterBFZ, SieBot, Nihil novi, Jccort, Arjen Dijks- man, Afernand74, Tesi1700, Melcombe, ImageRemovalBot, Kotniski, All Hallow’s Wraith, EoGuy, XPTO, Cp111, Masterpiece2000, No Free Nickname Left, SchreiberBike, Nafis ru, Chaosdruid, Dank, Frizchar, SilvonenBot, MystBot, Addbot, GargoyleBot, Metsavend, Favonian, Feketekave, NoEdward, Legobot, Luckas-bot, Yobot, Götz, ChristopheS, ArthurBot, LilHelpa, Xqbot, *feridiák, GrouchoBot, BigWednesday, False vacuum, Omnipaedista, Midgetman433, FrescoBot, Ironboy11, Saltus77, I dream of horses, Orenburg1, Double sharp, Trappist the monk, Hedviberit, TheAnkopinko15, Oracleofottawa, MicioGeremia, RjwilmsiBot, TjBot, In ictu oculi, John of Reading, WikitanvirBot, GA bot, 478jjjz, GoingBatty, Chimpionspeak, CrimsonBlue, A. Kupicki, Lateg, Bluesky10, Sergii.Fiot, Wing- man4l7, Puffin, Bomazi, ClueBot NG, Dr. Persi, Kstouras, Deer*lake, Pelle312, Helpful Pixie Bot, Jylothr, BG19bot, AvocatoBot, Met- ricopolus, Heathenreel, Wielki, BattyBot, Ninmacer20, ChrisGualtieri, Khazar2, Illia Connell, Dexbot, Mogism, Buspirtraz, VIAFbot, Jamesx12345, Chaim1995, Jbeck8924, Monochrome Monitor, Jossi2, Beleg00shyama, Eigenbra, K9re11, Monkbot, Knife-in-the-drawer and Anonymous: 125 • Stefan Banach Source: http://en.wikipedia.org/wiki/Stefan_Banach?oldid=660917996 Contributors: Kpjas, Zundark, Szopen, Piotr Gasiorowski, Andre Engels, Ben-Zin~enwiki, Edward, Michael Hardy, Cyde, Pagingmrherman, Vargenau, Charles Matthews, EALacey, Ark30inf, Jerzy, Jusjih, Flockmeal, Aleph4, Robbot, Jaredwf, MathMartin, Wickie, Halibutt, Ancheta Wis, Tosha, Giftlite, BenFrantz- Dale, MathKnight, Gadfium, Piotrus, Emax, Stako, Balcer, Almit39, PolishPoliticians, TonyW, Irpen, D6, EBL, Naive cynic, Maksym Ye., Bender235, Rgdboer, Robotje, Vegalabs, Darwinek, Mdd, Lysdexia, Eric Kvaalen, Arthena, Logologist, Japanese Searobin, Lem- berger 28, Isnow, Rjwilmsi, Koavf, Lockley, FlaBot, Volunteer Marek, Korg, Algebraist, YurikBot, Wavelength, RobotE, C777, Alex Bakharev, TheGrappler, Wiki alf, Rhythm, JohJak2, Molobo, Musashi miyamoto, Brian Tvedt, ArielGold, RG2, GrinBot~enwiki, Deuar, Yakudza, SmackBot, Selfworm, Jacek Kendysz, PeterSymonds, Xx236, DHN-bot~enwiki, AdamSmithee, OrphanBot, Vanished User 0001, Mhym, Khoikhoi, Spiritia, Tymek, Mathiasrex, Mgiganteus1, Grasyop, Kevin Murray, LessHeard vanU, INkubusse, JForget, James pic, Cydebot, Ntsimp, Galassi, DumbBOT, Thijs!bot, Escarbot, GiM, Salgueiro~enwiki, Matthew Fennell, RebelRobot, Beaumont, Waacstats, David Eppstein, Chris G, Pavlo Demchuk, S3000, CommonsDelinker, VirtualDelight, Silin2005, Salih, Januszek, Plinden- baum, JavierMC, VolkovBot, Sjones23, WarddrBOT, Nikita3, TXiKiBoT, Jmath666, PeterBFZ, Romuald Wróblewski, Ostap R, Nihil novi, Cmadame, Renatops, Monegasque, Polbot, Miyokan, ImageRemovalBot, Kotniski, ChandlerMapBot, Rockfang, Iohannes Animo- sus, Addbot, Bwallin, Dayewalker, Legobot, Luckas-bot, Yobot, AnomieBOT, ChristopheS, Materialscientist, Citation bot, Xqbot, Grou- choBot, Omnipaedista, Cristianrodenas, FrescoBot, Lothar von Richthofen, Citation bot 1, Chenopodiaceous, Kiefer.Wolfowitz, TobeBot, Trappist the monk, Hedviberit, Reaper Eternal, Ron asquith, Amyari, RjwilmsiBot, Stanislawow, Tommy2010, Demiurge1000, ClueBot NG, MelbourneStar, Deer*lake, Helpful Pixie Bot, Podgorec, ChrisGualtieri, Khazar2, VIAFbot, Vrave98, Oliszydlowski, Monkbot, KasparBot and Anonymous: 80 • Subset Source: http://en.wikipedia.org/wiki/Subset?oldid=658760669 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM, Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres, Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite, Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au- gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone, Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey, KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or- angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me, 188 CHAPTER 44. ZERMELO–FRAENKEL SET THEORY

345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot, .anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car- bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, Anonymous Dissident, James.Spudeman, PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63, Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97, Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar- ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char- vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot, Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski, AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23, Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17 and Anonymous: 179 • Transitive set Source: http://en.wikipedia.org/wiki/Transitive_set?oldid=659275410 Contributors: Edward, Charles Matthews, Jitse Niesen, Tobias Bergemann, Lethe, EmilJ, Oleg Alexandrov, Salix alba, Arthur Rubin, Mhss, Keithdunwoody, JRSpriggs, Vaughan Pratt, CBM, Gregbard, Roches, Ttwo, Franklin.vp, Addbot, Barak Sh, Luckas-bot, Xqbot, Erik9bot, Tkuvho, Wikielwikingo, EmausBot, ZéroBot, Deltahedron, Pastisch and Anonymous: 13 • Ultrafilter Source: http://en.wikipedia.org/wiki/Ultrafilter?oldid=659241903 Contributors: AxelBoldt, Zundark, Michael Hardy, Chinju, Stevan White, Charles Matthews, Timwi, Prumpf, MathMartin, Giftlite, Gene Ward Smith, Markus Krötzsch, Jason Quinn, Salasks, Paul August, EmilJ, TenOfAllTrades, Oleg Alexandrov, Graham87, Rjwilmsi, R.e.b., Vlad Patryshev, FlaBot, Karelj, YurikBot, Benja, Trovatore, Crasshopper, Arthur Rubin, Psolrzan, Eskimbot, Mhss, Henning Makholm, Physis, JRSpriggs, CRGreathouse, Gregbard, Nick Number, JAnDbot, .anacondabot, Magioladitis, EdwardLockhart, Sullivan.t.j, David Eppstein, Danimey, Quux0r, Gogobera, VolkovBot, Cbigorgne, M gol, Anchor Link Bot, Nsk92, Mpd1989, Hans Adler, Hugo Herbelin, Legobot, Luckas-bot, Yobot, Ht686rg90, Kilom691, AnomieBOT, Xqbot, Howard McCay, FrescoBot, Theorist2, Citation bot 1, Tkuvho, RjwilmsiBot, EmausBot, Wgunther, Bbbbbbbbba, Nosuchforever, CitationCleanerBot, Mark viking, Grabigail and Anonymous: 24 • Ultraproduct Source: http://en.wikipedia.org/wiki/Ultraproduct?oldid=664706565 Contributors: AxelBoldt, The Anome, Michael Hardy, Bcrowell, Jimfbleak, Dysprosia, Prumpf, Aleph4, Giftlite, Waltpohl, Gauge, Kwamikagami, EmilJ, Oleg Alexandrov, Mathbot, Yurik- Bot, Trovatore, SmackBot, Iridescent, Stotr~enwiki, Zero sharp, JRSpriggs, Alan R. Fisher, Gotozeus, TXiKiBoT, Rumping, Hans Adler, Klundarr, Addbot, Vevek, Tkuvho, Chricho, Tijfo098, GuySh, Nosuchforever, Spectral sequence, Monkbot and Anonymous: 11 • Uncountable set Source: http://en.wikipedia.org/wiki/Uncountable_set?oldid=664962517 Contributors: AxelBoldt, Tarquin, AstroNomer~enwiki, Taw, Toby Bartels, PierreAbbat, Patrick, Michael Hardy, Dominus, Kevin Baas, Revolver, Charles Matthews, Dysprosia, Hyacinth, Fi- bonacci, Aleph4, Robbot, Tobias Bergemann, Giftlite, Mshonle~enwiki, Fropuff, Noisy, Crunchy Frog, Func, Keenan Pepper, Oleg Alexandrov, Graham87, Island, Salix alba, FlaBot, Margosbot~enwiki, YurikBot, Piet Delport, Gaius Cornelius, Trovatore, Scs, Bota47, Arthur Rubin, Naught101, SmackBot, Bh3u4m, Bananabruno, SundarBot, Dreadstar, Germandemat, Loadmaster, Mets501, Limaner, Stephen B Streater, JRSpriggs, CRGreathouse, CBM, Gregbard, Thijs!bot, Dugwiki, Salgueiro~enwiki, JAnDbot, .anacondabot, Ttwo, Qatter, KarenJo90, SieBot, Phe-bot, ClueBot, Canopus1, DumZiBoT, Addbot, Yobot, Omnipaedista, BenzolBot, FoxBot, Vishwaraj.anand00, Mark viking, ILLUSION-ZONE and Anonymous: 33 • Universe (mathematics) Source: http://en.wikipedia.org/wiki/Universe_(mathematics)?oldid=634111780 Contributors: Mav, The Anome, Andre Engels, Toby Bartels, William Avery, Michael Hardy, Oliver Pereira, MartinHarper, Gabbe, Geoffrey~enwiki, Charles Matthews, Asar~enwiki, WhisperToMe, Jeoth, Robbot, Fredrik, Wile E. Heresiarch, Lethe, Waltpohl, Gdr, Discospinster, Rich Farmbrough, Paul August, Elwikipedista~enwiki, Mdd, Arthena, Linas, Isnow, Marudubshinki, Jshadias, Salix alba, John Baez, Mathbot, YurikBot, Zwobot, SmackBot, FlashSheridan, ArgentiumOutlaw, Acepectif, Lambiam, Scoty6776, Mets501, EdC~enwiki, Newone, CmdrObot, CBM, JAnDbot, Hut 8.5, Magioladitis, Zana Dark, David Eppstein, R'n'B, Huzzlet the bot, CooperDenn, Policron, Station1, Kumioko (re- named), Marino-slo, Hans Adler, DumZiBoT, Addbot, CarsracBot, Lightbot, Luckas-bot, Andy.melnikov, AnomieBOT, , XZeroBot, Erik9bot, MastiBot, Level777, ClueBot NG, BG19bot, Kephir, AdhesiveStation and Anonymous: 27 • Well-founded relation Source: http://en.wikipedia.org/wiki/Well-founded_relation?oldid=659840399 Contributors: The Anome, Ap, Michael Hardy, Dominus, TakuyaMurata, Cyp, Charles Matthews, VeryVerily, Aleph4, Mountain, Tobias Bergemann, Filemon, Marekpetrik, Lethe, Lupin, Waltpohl, Mani1, Paul August, EmilJ, Nahabedere, MZMcBride, R.e.b., FlaBot, Trovatore, Mikeblas, Crasshopper, Mar- lasdad, That Guy, From That Show!, SmackBot, Ron.garcia, Nbarth, Mmehdi.g, Mets501, CBM, WillowW, Pgagge, JAnDbot, Albmont, Leyo, Reedy Bot, Alexsmail, Thehotelambush, Mutilin, Addbot, KamikazeBot, RibotBOT, FrescoBot, Involutive-revolution, MerlIwBot, Wohlfundi, YiFeiBot, Some1Redirects4You and Anonymous: 22 • Zermelo–Fraenkel set theory Source: http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory?oldid=663078440 Con- tributors: AxelBoldt, Matthew Woodcraft, Zundark, Tarquin, Toby Bartels, Dwheeler, Patrick, Michael Hardy, MartinHarper, Bcrow- ell, Chinju, Haakon, Habj, Tim Retout, Schneelocke, Charles Matthews, Dcoetzee, Dysprosia, Greenrd, Hyacinth, VeryVerily, Fi- bonacci, JohnH~enwiki, Aleph4, Rursus, Tobias Bergemann, Giftlite, Smjg, Dratman, CyborgTosser, Mellum, Jorend, Ajgorhoe, Taran- toga~enwiki, David Sneek, Vsmith, Bender235, Elwikipedista~enwiki, Peter M Gerdes, Nortexoid, Obradovic Goran, Msh210, Su- ruena, TXlogic, Gible, Oleg Alexandrov, Joriki, OwenX, Drostie, Ma Baker, Hdante, Esben~enwiki, Dionyziz, MarSch, Salix alba, R.e.b., STarry, Chobot, Karch, YurikBot, Hairy Dude, Michael Slone, Piet Delport, Ksnortum, Ogai, Trovatore, Twin Bird, Expen- sivehat, Insipid, Jpbowen, Crasshopper, Wknight94, Arthur Rubin, Josh3580, Banus, Otto ter Haar, Schizobullet, A bit iffy, SmackBot, Fulldecent, Mhss, Darth Panda, Foxjwill, Tsca.bot, Miguel1626, TKD, Allan McInnes, Grover cleveland, Acepectif, Jon Awbrey, Meni Rosenfeld, Stefano85, Vina-iwbot~enwiki, Noegenesis, Rainwarrior, Dicklyon, Tophtucker, JRSpriggs, CRGreathouse, CBM, Myasuda, Gregbard, Awmorp, Thijs!bot, Whooooooknows, Odoncaoa, Jirka6, VictorAnyakin, JAnDbot, Quentar~enwiki, Giler, Mathfreq, Omi- cron18, JustinRosenstein, Diroth, The Real Marauder, Numbo3, Ttwo, Trumpet marietta 45750, Policron, JavierMC, The enemies of god, Crisperdue, Pasixxxx, Magmi, Bistromathic, Henry Delforn (old), Jjepfl, C xong, JP.Martin-Flatin, Alexbot, Iohannes Animosus, Palnot, Marc van Leeuwen, Addbot, Matěj Grabovský, Yobot, AnomieBOT, Materialscientist, La comadreja, Control.valve, Vladimir- Reshetnikov, Nicolas Perrault III, Andrewjameskirk, NSH002, Tkuvho, Zdorovo, ClueBot NG, Chetrasho, Snotbot, Helpful Pixie Bot, Brad7777, Daysrr, Khazar2, Jochen Burghardt, Mark viking and Anonymous: 117

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