DOCSLIB.ORG

Well Ordered Sets (Continued) Theorem 8 Given Any Two Well-Ordered Sets, Either They Are Isomorphic, Or One Is Isomorphic to an Initial Segment of the Other

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Well Ordered Sets (Continued) Theorem 8 Given Any Two Well-Ordered Sets, Either They Are Isomorphic, Or One Is Isomorphic to an Initial Segment of the Other Well Ordered Sets (continued) Theorem 8 Given any two well-ordered sets, either they are isomorphic, or one is isomorphic to an initial segment of the other. Proof Let ha, < i and hb, ⊳i be well-ordered sets. Let d be such that d∈ / b (take e.g. d = b). Define f : a → b ∪ { d} to satisfy the ⊳−least element in b\Rng (f ↾a ) if b\Rng (f ↾a ) 6= ∅ f(x) = x x ½ d otherwise (12) The existence of such f is guaranteed by the previous theorem 1 since for each x from the well ordered set a, f(x) is uniquely determined in terms of x and f ↾ax. Observe that if x1, x 2 ∈ a and x1 < x 2 then ax1 ⊂ ax2 so b\Rng (f ↾ax2 ) ⊆ b\Rng (f ↾ax1 ) and hence we have either both f(x1) and f(x2) are in b and f(x1) ⊳ f(x2), (f(x2) ∈ b\Rng (f ↾ax2 ) and f(x1) ∈ Rng (f ↾ax2 ) so f(x1) 6= f(x2)) or only f(x1) is in b and f(x2) = d, or f(x1) = f(x2) = d. Define g = f ∩ (a × b), i.e. g(x) = f(x) except when f(x) = d in which case x∈ / Dom (g). Let s = Dom (g), t = Rng (g). If x1, x 2 ∈ a, x2 ∈ s and x1 < x 2 then x1 ∈ s by the above observation. Also, if y1, y 2 ∈ b, y2 ∈ t and y1 ⊳ y2 then y1 ∈ t since if y2 = f(x) for some x in a then y1 must be in Rng (f ↾ax). It follows by Example 6.2 that s = a or s = ax for some x ∈ a, and t = b or t = by for some y ∈ b. Note that s is well ordered by the restriction of < to s and t is well ordered by the restriction of ⊳ to t. g : s → t is a bijection since it is surjective by definition of t and if x1, x 2 ∈ s, x1 6= x2 then assuming without loss of generality that x1 < x 2 we have f(x1) ⊳ f(x2) by the above observation, so g an order preserving bijection and as such an isomorphism between s and t by Corollary 3. 1For θ(x, u, y ) we can take e.g. the formula [x ∈ a & F n (u) & Dom (u) = ax & b\Rng (u) 6= ∅ & y = Lst (b\Rng (u))] ∨ [¬(x ∈ a & F n (u) & Dom (u) = ax & b\Rng (u) 6= ∅) & y = d] where y = Lst (w) stands for y ∈ w & ( ∀z ∈ w) y E z. It is easy to check that ∀x∀u∃!yθ (x, u, y ), and that for x ∈ a, (12) corresponds to θ(x, f ↾ax, f (x)). 48 Since s = a or s = ax for some x ∈ a, and t = b or t = by for some y ∈ b, and since the case of both s = ax and y = by cannot occur (it would mean f(x) = d but b\Rng (f ↾ax) 6= ∅), the theorem follows. ¥ Corollary 4 If sets a and b can be well ordered then either a ¹ b or b ¹ a. Consequently, if all sets can be well ordered then for any two sets a, b we have a ¹ b or b ¹ a. Later, we will show the converse: if a ¹ b or b ¹ a for any two sets a, b, then every set can be well ordered! The statement that all sets can be well ordered is usually referred to as the Well Ordering Theorem (WOT) and the statement that for any two sets a, b we have a ¹ b or b ¹ a is referred to as the Law of Trichotomy (LOT). Hence we have shown that it follows from out axioms that WOT implies LOT and we will soon show that is also follows that LOT implies WOT. (However, neither can be proved in ZF. We will return to this later.) Proposition 45 (i) Let a be a set well-ordered by ⊳ and let f : a ¹ a be order-preserving. Then x E f(x) for each x ∈ a. (ii) There can be no order-preserving injective function from a well ordered set to an initial segment of it. Proof (i) Assume that the statement does not hold and let c = {x ∈ a | f(x) ⊳ x}. Note that c is non-empty since for some x ∈ a, x E f(x) does not hold, and ⊳ is total. Let x0 be the least element of c. We have f(x0) ⊳ x0 so since f is injective and order preserving also f(f(x0)) ⊳ f(x0) but that means f(x0) ∈ c and f(x0) ⊳ x0, contradicting the choice of x0 as the least element in c. (ii) Let a be well ordered by ⊳, x ∈ a. If f was an order preserving injective function from a to ax then f(x) ⊳ x which contradicts (i). ¥ Corollary 5 The isomorphism in Theorem 8 is unique. Proof See Example 6.4. Ordinals Well-ordered sets generalize the notion of counting as represented by natural numbers, when counting is understood as arranging elements in order, starting from a first one and proceeding in discrete steps. Assuming again we have N for the moment, note that for any finite set there is (up to isomorphism) only one way to count it: if n ∈ N and a set a has n elements and ha, ⊳i is well ordered then ha, ⊳i is isomorphic to hNn <i byTheorem 8, and because finite sets can only be equinumerous if they have the same number 49 of elements. However, we have seen that on N there can be essentially different well-orderings (cf. Example 5.4(ii)). Non-isomorphic well-orderings correspond to essentially different outcomes we may get when counting sets. Isomorphic well orderings share an important feature, similarly to equinumerous sets having the same cardinality. We say that isomorphic well ordered sets have the same order type. We could proceed similarly as before and talk about an order type being that which is shared by isomorphic well-ordered sets. However, we are now in a position to say exactly what we mean, and define a unique set representing each such class of isomorphic well orderings. When we have done it, we will proceed to use it to define what we mean by cardinal numbers, too. Definition 20 Let a be a set. Then ∈a is the relation on a defined by ∈a= {h x, y i | x, y ∈ a & x ∈ y} . ∈a is irreflexive on a by virtue of the Axiom of Foundation, but in general it is not transitive nor total. We shall be interested in special sets which are totally ordered by ∈a. First we point out some properties of sets a totally ordered by ∈a. Lemma 9 Let a be a set such that ∈a totally orders a, and let b ⊆ a and w ∈ b. Then w is the least element of b (with respect to ∈a) if and only if w ∩ b = ∅. Proof w is the least element of b just when ( ∀y ∈ b) ( y = w ∨ w ∈a y) which, by virtue of ∈a being total, is equivalent to any of the following (∀y ∈ b) ¬y ∈a w ⇐⇒ (∀y ∈ b) ¬y ∈ w ⇐⇒ w ∩ b = ∅. ¥ Corollary 6 A set a is totally ordered by ∈a if and only if it is well ordered by it. Proof This follows by the above Lemma and the Axiom of Foundation. ¥ Corollary 7 A set a is totally ordered by ∈a if and only if For all x, y ∈ a, exactly one of x = y, x ∈ y, y ∈ x holds . (OT ) Proof By the previous Corollary and Proposition 42 (since ( OT ) implies the existence of a least elements for any non-empty set subset of a, as in the proof of Lemma 9). Definition 21 A set a is transitive if it contains all elements of its elements, that is, (∀x ∈ a) x ⊆ a. 50 Warning The terminology is established but potentially confusing: to say that a relation is transitive is not the same thing as saying that a set is transitive (even though any relation is also a set!). It is always clear from the context what is meant. Definition 22 A set a is an ordinal if it is transitive and totally ordered by ∈a= {h x, y i | x, y ∈ a & x ∈ y} Proposition 46 If a is an ordinal then a is well ordered by ∈a. Proof From Corollary 6. ¥ We remark that when Axiom of Foundation is not assumed, an ordinal is usually defined as a transitive set a which is well ordered by ∈a. As Proposition 46 shows, in ZF this is equivalent to our definition. Proposition 47 A set a is an ordinal if and only if it is transitive and For all x, y ∈ a, exactly one of x = y, x ∈ y, y ∈ x holds . (OT ) Proof From Corollary 7. ¥ For example, ∅, {∅} , {∅ , {∅}} , {∅ , {∅} , {∅ , {∅}}} are ordinals. We will use lower case Greek letters α,β,γ,δ,... for ordinals. Let Ord stand for the class of all ordinals. We will write Ord (x) to mean x is an ordinal. We will need the following properties of ordinals. Recall that x ⊂ y stands for x ⊆ y and x 6= y. Proposition 48 Let α, β be ordinals, d a set. (i) If d ∈ α then d ⊂ α. (ii) If d ∈ α then d ∈ Ord . (iii) β ∈ α ⇐⇒ β ⊂ α. (iv) Exactly one of α ∈ β, α = β, β ∈ α holds. Proof (i) By transitivity of α, if d ∈ α then d ⊆ α.
Load more

Recommended publications
  • Order Types and Structure of Orders
    ORDER TYPES AND STRUCTURE OF ORDERS BY ANDRE GLEYZALp) 1. Introduction. This paper is concerned with operations on order types or order properties a and the construction of order types related to a. The reference throughout is to simply or linearly ordered sets, and we shall speak of a as either property or type. Let a and ß be any two order types. An order A will be said to be of type aß if it is the sum of /3-orders (orders of type ß) over an a-order; i.e., if A permits of decomposition into nonoverlapping seg- ments each of order type ß, the segments themselves forming an order of type a. We have thus associated with every pair of order types a and ß the product order type aß. The definition of product for order types automatically associates with every order type a the order types aa = a2, aa2 = a3, ■ ■ ■ . We may further- more define, for all ordinals X, a Xth power of a, a\ and finally a limit order type a1. This order type has certain interesting properties. It has closure with respect to the product operation, for the sum of ar-orders over an a7-order is an a'-order, i.e., a'al = aI. For this reason we call a1 iterative. In general, we term an order type ß having the property that ßß = ß iterative, a1 has the following postulational identification: 1. a7 is a supertype of a; that is to say, all a-orders are a7-orders. 2. a1 is iterative.
    [Show full text]
  • Handout from Today's Lecture
    MA532 Lecture Timothy Kohl Boston University April 23, 2020 Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 1 / 26 Cardinal Arithmetic Recall that one may define addition and multiplication of ordinals α = ot(A, A) β = ot(B, B ) α + β and α · β by constructing order relations on A ∪ B and B × A. For cardinal numbers the foundations are somewhat similar, but also somewhat simpler since one need not refer to orderings. Definition For sets A, B where |A| = α and |B| = β then α + β = |(A × {0}) ∪ (B × {1})|. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 2 / 26 The curious part of the definition is the two sets A × {0} and B × {1} which can be viewed as subsets of the direct product (A ∪ B) × {0, 1} which basically allows us to add |A| and |B|, in particular since, in the usual formula for the size of the union of two sets |A ∪ B| = |A| + |B| − |A ∩ B| which in this case is bypassed since, by construction, (A × {0}) ∩ (B × {1})= ∅ regardless of the nature of A ∩ B. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 3 / 26 Definition For sets A, B where |A| = α and |B| = β then α · β = |A × B|. One immediate consequence of these definitions is the following. Proposition If m, n are finite ordinals, then as cardinals one has |m| + |n| = |m + n|, (where the addition on the right is ordinal addition in ω) meaning that ordinal addition and cardinal addition agree. Proof. The simplest proof of this is to define a bijection f : (m × {0}) ∪ (n × {1}) → m + n by f (hr, 0i)= r for r ∈ m and f (hs, 1i)= m + s for s ∈ n.
    [Show full text]
  • Types for Describing Coordinated Data Structures
    Types for Describing Coordinated Data Structures Michael F. Ringenburg∗ Dan Grossman [email protected] [email protected] Dept. of Computer Science & Engineering University of Washington, Seattle, WA 98195 ABSTRACT the n-th function). This section motivates why such invari- Coordinated data structures are sets of (perhaps unbounded) ants are important, explores why the scope of type variables data structures where the nodes of each structure may share makes the problem appear daunting, and previews the rest abstract types with the corresponding nodes of the other of the paper. structures. For example, consider a list of arguments, and 1.1 Low-Level Type Systems a separate list of functions, where the n-th function of the Recent years have witnessed substantial work on powerful second list should be applied only to the n-th argument of type systems for safe, low-level languages. Standard moti- the first list. We can guarantee that this invariant is obeyed vation for such systems includes compiler debugging (gener- by coordinating the two lists, such that the type of the n-th ated code that does not type check implies a compiler error), argument is existentially quantified and identical to the ar- proof-carrying code (the type system encodes a safety prop- gument type of the n-th function. In this paper, we describe erty that the type-checker verifies), automated optimization a minimal set of features sufficient for a type system to sup- (an optimizer can exploit the type information), and manual port coordinated data structures. We also demonstrate that optimization (humans can use idioms unavailable in higher- two known type systems (Crary and Weirich’s LX [6] and level languages without sacrificing safety).
    [Show full text]
  • Dimensional Functions Over Partially Ordered Sets
    Intro Dimensional functions over posets Application I Application II Dimensional functions over partially ordered sets V.N.Remeslennikov, E. Frenkel May 30, 2013 1 / 38 Intro Dimensional functions over posets Application I Application II Plan The notion of a dimensional function over a partially ordered set was introduced by V. N. Remeslennikov in 2012. Outline of the talk: Part I. Definition and fundamental results on dimensional functions, (based on the paper of V. N. Remeslennikov and A. N. Rybalov “Dimensional functions over posets”); Part II. 1st application: Definition of dimension for arbitrary algebraic systems; Part III. 2nd application: Definition of dimension for regular subsets of free groups (L. Frenkel and V. N. Remeslennikov “Dimensional functions for regular subsets of free groups”, work in progress). 2 / 38 Intro Dimensional functions over posets Application I Application II Partially ordered sets Definition A partial order is a binary relation ≤ over a set M such that ∀a ∈ M a ≤ a (reflexivity); ∀a, b ∈ M a ≤ b and b ≤ a implies a = b (antisymmetry); ∀a, b, c ∈ M a ≤ b and b ≤ c implies a ≤ c (transitivity). Definition A set M with a partial order is called a partially ordered set (poset). 3 / 38 Intro Dimensional functions over posets Application I Application II Linearly ordered abelian groups Definition A set A equipped with addition + and a linear order ≤ is called linearly ordered abelian group if 1. A, + is an abelian group; 2. A, ≤ is a linearly ordered set; 3. ∀a, b, c ∈ A a ≤ b implies a + c ≤ b + c. Definition The semigroup A+ of all nonnegative elements of A is defined by A+ = {a ∈ A | 0 ≤ a}.
    [Show full text]
  • Generic Haskell: Applications
    Generic Haskell: applications Ralf Hinze1 and Johan Jeuring2,3 1 Institut f¨urInformatik III, Universit¨atBonn R¨omerstraße 164, 53117 Bonn, Germany [email protected] http://www.informatik.uni-bonn.de/~ralf/ 2 Institute of Information and Computing Sciences, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, The Netherlands [email protected] http://www.cs.uu.nl/~johanj/ 3 Open University, Heerlen, The Netherlands Abstract. 1 Generic Haskell is an extension of Haskell that supports the construc- tion of generic programs. This article describes generic programming in practice. It discusses three advanced generic programming applications: generic dictionaries, compressing XML documents, and the zipper. When describing and implementing these examples, we will encounter some advanced features of Generic Haskell, such as type-indexed data types, dependencies between and generic abstractions of generic functions, ad- justing a generic function using a default case, and generic functions with a special case for a particular constructor. 1 Introduction A polytypic (or generic, type-indexed) function is a function that can be instan- tiated on many data types to obtain data type specific functionality. Examples of polytypic functions are the functions that can be derived in Haskell [50], such as show, read, and ‘ ’. In [23] we have introduced type-indexed functions, and we have shown how to implement them in Generic Haskell [7]. For an older introduction to generic programming, see Backhouse et al [4]. Why is generic programming important? Generic programming makes pro- grams easier to write: – Programs that could only be written in an untyped style can now be written in a language with types.
    [Show full text]
  • Mouse Pairs and Suslin Cardinals
    Mouse pairs and Suslin cardinals John R. Steel October 2018 Abstract Building on the results of [13] and [12], we obtain optimal Suslin representations for M mouse pairs. This leads us to an analysis of HOD , for any model M of ADR in which there are Wadge-cofinally many mouse pairs. We show also that if there is a least branch hod pair with a Woodin limit of Woodin cardinals, then there is a model of AD+ in which not every set of reals is ordinal definable from a countable sequence of ordinals. 0 Introduction The paper [13] introduces mouse pairs, and develops their basic theory. It then uses one variety of mouse pair, the least branch hod pairs, to analyze HOD in certain models of the Axiom of Determinacy. Here we shall carry that work further, by obtaining optimal Suslin representations for mouse pairs. This enables us to show to show that the collection of models M of ADR to which the HOD-analysis of [13] applies is closed downward under Wadge reducibility. It also leads to a characterization of the Solovay sequence in such models M of ADR in terms of the Woodin cardinals of HODM . Unless otherwise stated, we shall assume AD+ throughout this paper. 0.1 Mouse pairs Our intended reader has already read [13], but let us recall, in outline, some of the definitions and results of that paper. _ E~ ~ ~ A pure extender premouse is a transitive structure M = (Jα ; 2; E; F ), where E F is a M coherent sequence of extenders. (F = ; is allowed.) We use Jensen indexing: if E = Eα , then +;M α = iE(crit(E) ).
    [Show full text]
  • Omega-Models of Finite Set Theory
    ω-MODELS OF FINITE SET THEORY ALI ENAYAT, JAMES H. SCHMERL, AND ALBERT VISSER Abstract. Finite set theory, here denoted ZFfin, is the theory ob- tained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) em- ployed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In par- ticular, we present a new method for building ω-models of ZFfin that leads to a perspicuous construction of recursive nonstandard ω-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an ω-model. The central theorem of the paper is the following: Theorem A. For every graph (A, F ), where F is a set of un- ordered pairs of A, there is an ω-model M of ZFfin whose universe contains A and which satisfies the following conditions: (1) (A, F ) is definable in M; (2) Every element of M is definable in (M, a)a∈A; (3) If (A, F ) is pointwise definable, then so is M; (4) Aut(M) =∼ Aut(A, F ). Theorem A enables us to build a variety of ω-models with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin.
    [Show full text]
  • Chapter 8 Ordered Sets
    Chapter VIII Ordered Sets, Ordinals and Transfinite Methods 1. Introduction In this chapter, we will look at certain kinds of ordered sets. If a set \ is ordered in a reasonable way, then there is a natural way to define an “order topology” on \. Most interesting (for our purposes) will be ordered sets that satisfy a very strong ordering condition: that every nonempty subset contains a smallest element. Such sets are called well-ordered. The most familiar example of a well-ordered set is and it is the well-ordering property that lets us do mathematical induction in In this chapter we will see “longer” well ordered sets and these will give us a new proof method called “transfinite induction.” But we begin with something simpler. 2. Partially Ordered Sets Recall that a relation V\ on a set is a subset of \‚\ (see Definition I.5.2 ). If ÐBßCÑ−V, we write BVCÞ An “order” on a set \ is refers to a relation on \ that satisfies some additional conditions. Order relations are usually denoted by symbols such asŸ¡ß£ , , or . Definition 2.1 A relation V\ on is called: transitive ifÀ a +ß ,ß - − \ Ð+V, and ,V-Ñ Ê +V-Þ reflexive ifÀa+−\+V+ antisymmetric ifÀ a +ß , − \ Ð+V, and ,V+ Ñ Ê Ð+ œ ,Ñ symmetric ifÀ a +ß , − \ +V, Í ,V+ (that is, the set V is “symmetric” with respect to thediagonal ? œÖÐBßBÑÀB−\ש\‚\). Example 2.2 1) The relation “œ\ ” on a set is transitive, reflexive, symmetric, and antisymmetric. Viewed as a subset of \‚\, the relation “ œ ” is the diagonal set ? œÖÐBßBÑÀB−\×Þ 2) In ‘, the usual order relation is transitive and antisymmetric, but not reflexive or symmetric.
    [Show full text]
  • Order on Order Types∗
    Order on Order Types∗ Alexander Pilz1 and Emo Welzl2 1 Institute for Software Technology, Graz University of Technology, Austria [email protected] 2 Department of Computer Science, Institute of Theoretical Computer Science ETH Zürich, Switzerland [email protected] Abstract Given P and P 0, equally sized planar point sets in general position, we call a bijection from P to P 0 crossing-preserving if crossings of connecting segments in P are preserved in P 0 (extra crossings may occur in P 0). If such a mapping exists, we say that P 0 crossing-dominates P , and if such a mapping exists in both directions, P and P 0 are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.) We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.
    [Show full text]
  • General-Well-Ordered Sets
    J. Austral. Math. Soc. 19 (Series A), (1975), 7-20. GENERAL-WELL-ORDERED SETS J. L. HICKMAN (Received 10 April 1972) Communicated by J. N. Crossley It is of course well known that within the framework of any reasonable set theory whose axioms include that of choice, we can characterize well-orderings in two different ways: (1) a total order for which every nonempty subset has a minimal element; (2) a total order in which there are no infinite descending chains. Now the theory of well-ordered sets and their ordinals that is expounded in various texts takes as its definition characterization (1) above; in this paper we commence an investigation into the corresponding theory that takes character- ization (2) as its starting point. Naturally if we are to obtain any differences at all, we must exclude the axiom of choice from our set theory. Thus we state right at the outset that we are working in Zermelo-Fraenkel set theory without choice. This is not enough, however, for it may turn out that the axiom of choice is not really necessary for the equivalence of (1) and (2), and if this were so, we would still not obtain any deviation from the "classical" theory. Thus the first section of this paper will be devoted to showing that we can in fact have sets that are well-ordered according to (2) but not according to (1). Naturally we have no hope of reversing this situation, for it is a ZF-theorem that any set well-ordered by (1) is also well-ordered by (2).
    [Show full text]
  • An Introduction to Ordinals
    An Introduction to Ordinals D. Salgado, N. Singer #blairlogicmath November 2017 D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 1 / 39 Introduction D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 2 / 39 History Bolzano defined a set as \an aggregate so conceived that it is indifferent to the arrangement of its members" (1883) Cantor defined set membership, subsets, powersets, unions, intersections, complements, etc. Given two sets A and B, Cantor says they have the same cardinality iff there exists a bijection between them Denoted jAj = jBj More generally, jAj ≤ jBj iff there is an injection from A to B Every set has a cardinal number, which represents its cardinality @0 Infinite sets have cardinalities @0; 2 ;::: Arithmetic can be defined, e.g. jAj + jBj = j(A × f0g) [ (B × f1g)j D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 3 / 39 That structure is derived from the order of the set The notions of cardinality only apply to unstructured sets and are determined by bijections Every set also has an order type determined by order-preserving bijections: f : A ! B is a bijection and 8x; y 2 A [(x < y) ! (f (x) < f (y))] Intuition Cantor viewed sets as fundamentally structured When we view the natural numbers, why does it make sense to think of them as just an “infinite bag of numbers"? The natural numbers is constructed in an inherently structured way: through successors D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November
    [Show full text]
  • MA532 Lecture
    MA532 Lecture Timothy Kohl Boston University April 9, 2020 Timothy Kohl (Boston University) MA532 Lecture April 9, 2020 1 / 24 Order Types and Order Type Arithmetic Back in calculus, one of the points that was frequently made was that ’∞’ is not a number! This point is made, in particular, to prevent one from trying to (carelessly) try to make sense of statements like ’∞ +1= ∞’ which of course lead to the non-sensical conclusion that 0 = 1. In the setting of the natural numbers this would be translated into the statement that ’ω’ isn’t a number, but can we make sense of ω +1 in any way? However, we will see that one can make sense of expressions like ’1 + ω’ and ’ω +1’ which is a formal kind of arithmetic, but does not imply non-sensical statements like ’1 = 0’. Timothy Kohl (Boston University) MA532 Lecture April 9, 2020 2 / 24 Before we get to this ordinal (actually order type) arithmetic, we need to define what are known as order types, which effectively allow us to distinguish between different order relations. Recall the following. Definition Let (A, A) and (B, B ) be ordered sets then (A, A) and (B, B ) are said to be order isomorphic if there is a bijection f : A → B such that x A y implies f (x) B f (y), and we write (A, A) =∼ (B, B ) to denoted this isomorphism. Timothy Kohl (Boston University) MA532 Lecture April 9, 2020 3 / 24 For a finite set A with n elements, there are n! rearrangements (permutations) of A and each such permutation is representable by a bijection σ : A → A.
    [Show full text]