The of cardinal

We wish to extend the familiar operations of , and expo- nentiation of the natural numbers so as to apply to all cardinal numbers. We start with addition and multiplication, then go on consider the unary exponen- tiation assigning 2 n to each n and finally the general binary assigning mn to numbers n, m .

Addition and Multiplication Lemma 2 For finite sets A, B , (i) If A ∩ B = ∅ then A ∪ B is finite and |A ∪ B| = |A| + |B|, (ii) A × B is finite and |A × B| = |A| · | B|.

Proof see Example 1.9. For sets A, B let A + B = ( {0} × A) ∪ ({1} × B). We call this the disjoint of A and B.

Lemma 3 If A ∼ C and B ∼ D then (i) A + B ∼ C + D, (ii) A × B ∼ C × D.

Proof Let f : A ∼ C, g : B ∼ D. Then h : A + B → C + D and e : A × B → C × D defined by

h(0 , a ) = (0 , f (a)) for a ∈ A, h (1 , b ) = (1 , g (b)) for b ∈ B

e(a, b ) = ( f(a), g (b)) for a ∈ A, b ∈ B are so the result follows.

Definition 10 For cardinal numbers n, m let (i) n + m = |A + B| for some/any sets A, B such that n = |A| and m = |B|, (ii) n · m = |A × B| for some/any sets A, B such that n = |A| and m = |B|,

Note that the previous lemma ensures that these definitions make sense.

Proposition 21 For cardinal numbers n, m and any sets A, B such that A ∩ B = ∅ and n = |A| and m = |B| we have n + m = |A ∪ B|.

Proof For A, B we have the obvious h : A + B ∼ A ∪ B defined by h(0 , a ) = a for a ∈ A and h(1 , b ) = b for b ∈ B.

21 Proposition 22 Let n, p, m, k be cardinal numbers such that n ≤ p and m ≤ k. Then (i) n + m ≤ p + k, (ii) n · m ≤ p · k. Proof Let A, B, C, D be some sets with n = |A|, m = |B|, p = |C|, k = |D| and let f : A ¹ C, g : B ¹ D. The function h and e defined as in the proof of Lemma 3 are injections from A + B to C + D and from A × B to C × D respectively, so the result follows. Proposition 23 If m, n, k are cardinal numbers then (i) (m + n) + k = m + ( n + k),

(ii) m + n = n + m, (iii) m · (n + k) = m · n + m · k, (iv) (m · n) · k = m · (n · k), (v) m · n = n · m. Proof We will prove (i), leaving the rest as exercise. Let A, B, C be some pairwise disjoint sets with m = |A|, n = |B|, k = |C|. By Proposition 21 we have m + n = |A ∪ B|, n + k = |B ∪ C| so since ( A ∪ B) ∩ C = ∅ and A ∩ (B ∪ C) = ∅, we only need to show that ( A ∪ B) ∪ C ∼ A ∪ (B ∪ C) which is clearly true since the sets are equal.

Unary Exponentiation Lemma 4 For each finite A, the set P ow (A) is finite and |P ow (A)| = 2 |A|. Proof see Example 1.9. Lemma 5 If A ∼ B then P ow (A) ∼ P ow (B). Proof Let f : A ∼ B. Then h : P ow (A) → P ow (B) defined by h(X) = {f(x) | x ∈ X} for X ∈ P ow (A) is easily checked to be a bijection. Definition 11 For each n let 2n = |P ow (A)| for some/any set A such that n = |A|.

22 Proposition 24 For cardinal numbers n, m (i) If n ≤ m then 2n ≤ 2m, (ii) 2n+m = 2 n · 2m, (iii) n < 2n, (iv) c = 2 ℵ0 .

Proof (i) is an exercise. (ii) Let A and B be disjoint sets with n and m respectively. The mapping h : P ow (A) × P ow (B) → P ow (A ∪ B) defined by h(X, Y ) = X ∪ Y is a bijection, hence 2 n · 2m = 2 n+m. (iii) is the Cantor’s theorem. (iv) is the Proposition 19.

Binary Exponentiation Recall that for sets A, B we defined AB = {f | f : B→A}.

Lemma 6 If A, B are finite sets then so is AB and |AB| = |A||B|.

Proof See Example 3.1.

Lemma 7 If A ∼ C and B ∼ D then AB ∼ CD.

Proof Let f : A ∼ C, g : B ∼ D. Then h : AB → CD defined by

h(e) = f ◦ (e ◦ g−1) for e ∈ AB is a bijection. (Draw a picture.)

Definition 12 For cardinal numbers n, m let

mn = |AB| for some/any sets A, B such that m = |A| and n = |B|.

Note that for any set A, A∅ contains exactly one (the function that does not assign anything to anything), hence n0 = 1 for every cardinal number n. If A is non-empty then ∅A is empty so for n 6= 0 we have 0 n = 0. The following result ensures that binary exponentiation agrees with unary ex- ponentiation.

Proposition 25 For any set A, P ow (A) ∼ { 0, 1}A.

23 Proof The function that assigns Z ⊆ A its characteristic function χZ : A → { 0, 1}

1 if a ∈ Z χZ (a) = ½ 0 if a∈ / Z is a bijection.

Proposition 26 For cardinal numbers n, m, p, k ,

(i) If 0 < n and m ≤ k then nm ≤ nk, (ii) If n ≤ p then nm ≤ p m.

Proof Exercise. Proposition 27 If m, n, k are cardinal numbers then

(i) mn+k = mn · mk, (ii) mn·k = ( mn)k,

(iii) (m · n)k = mk · nk. Proof Let A, B, C be sets with |A| = m, |B| = n, |C| = k and B ∩ C = ∅. (i) The function h : AB∪C → (AB) × (AC ) which assigns e ∈ AB∪C the pair hf, g i with f being the restriction of e to B and g being the restriction of e to C is a bijection. (ii) The function h : AB×C → (AB)C which assigns e ∈ AB×C the function f ∈ (AB)C satisfying

[f(c)]( b) = e(b, c ) for c ∈ C and b ∈ B is a bijection. (iii) The function h : ( A × B)C → (AC ) × (BC ) which assigns e ∈ (A × B)C the pair hf, g i such that e(c) = hf(c), g (c)i for c ∈ C is a bijection.

24 Ordinal Numbers Cardinal numbers generalize the natural numbers in their capacity of measuring sizes of sets, telling us how many elements a set has. However, natural numbers are also used for some things one by one, like putting elements of a set in order or labeling steps in some process. The distinction is perhaps subtle because in the case of finite sets, whichever way we order the elements of a set one after another, we get a finite sequence of the same length, and this length is equal to the number of elements in the finite set. Cantor was led to consider this role of natural numbers early on, when he introduced the notion of the derived set for a given of R: the derived set of a set A ⊆ R is the set of its points, that is, A without its isolated points. Cantor was interested in what happens when we start with a set and repeat the operation of taking the derived set. There are sets A such that if we keep taking the derived set we obtain a sequence A′, (A′)′,... of ever smaller sets such that the intersection of all of them again has isolated points so we may take the derived set of that again and so on .... For steps in this process Cantor needed a sequence of ordinal numbers which start just as the natural numbers but continue:

0, 1, 2, 3,...,ω, ω +1 , ω +2 ,...,ω + ω, ω ·2+1 ,...,ω ·3, ω ·3+1 ,...,...,ω ·ω,... ω·2 | {z } Cantor called numbers like these ordinal numbers 1. Note that when consid- ering e.g. the set {0, 1, 2,...,ω,ω + 1 ,...,ω + ω} we find that it is equinu- merous with N, although counting one by one up to ω + ω is ‘much longer’ than counting up to ω. Cantor developed the arithmetic of both cardinal and ordinal numbers and they are quite different . Both cardinal and ordinal addi- tion/multiplication/exponentiation generalize the corresponding operations on natural numbers and the same notation (+ , · , something something ) is used for both 2. We will not talk about the arithmetic of ordinal numbers until much later on in the course.

1We will define them precisely later. 2So the meaning has to be understood from the context.

25 Paradoxes

We have developed some naive , largely as it was first developed, as a framework accomodating previously known and studied mathematical struc- tures like natural, rational and real numbers. A framework which allowed treat- ing these collections of numbers along with their arbitrary subcollections and collections of their subcollections as a single sort of objects, namely sets. There was a considerable resistance to this process which we have already men- tioned, notably from Leopold Kronecker, who opposed even the publication of Cantor’s papers and his obtaining a mathematical position at Gottingen or Berlin University. On the other hand the freedom afforded by this new frame- work appealed to many. In 1900 declared the set theory to be a genuine part of mathematics, asserting that ”no one can drive us out of the paradise created for us by Cantor” and making the proof or refutation of the Hypothesis the first on the list of his famous problems. At the turn of the century however, paradoxes were surfacing which appeared to vindicate the sceptics. It seemed as if the paradoxes of the infinite were in no way resolved by the new theory but merely pushed a bit further: what is the of the set V of all sets? V must contain all its , so P ow (V ) ¹ V and hence |P ow (V )| ≤ | V | but a diagonal argument (Cantor’s theorem) shows this to be absurd. Cantor’s solution 3 was to say that some collections are too big to be sets. Hence the set of all sets is not a set but an inconsistent totality which we cannot comprehend. Such limitation of size can also deal with the most notorious paradox of the time, Russell’s Paradox (1902): Let M be the set of all sets that are not their own element, M = {x | x∈ / x}. Then M ∈ M ⇐⇒ M∈ / M which is clearly absurd. However, other problems could not be resolved on the grounds that the sets involved are too big, like Berry’s Paradox (1906): let n be the least that cannot be described in English by less than 100 words. Such a number clearly exists as the least number in the non- of natural numbers that cannot be described in English by less than 100 words (there are only finitely many descriptions with less than 100 words) but we have just described it by less than 100 words.

Some saw this as a confirmation that set theory does not make sense. In 1908, Henri Poincare referred to set theory as something future generation will see as a disease from which mathematics has recovered ... However, a lot of mathematics

3Which he did not arrive at lightly: all his mathematics was motivated and linked to theology and denying the possibility of existence of the set of all sets appeared to limit God’s power. Eventually, Cantor came to talk about two sorts of actual infinity, the absolute infinity that only pertains to God’s existence and is incomprehensible for mankind, and the transfinite which can be understood by us.

26 was already using set theory, like Cantor’s own investigations of real numbers, Lebesgue’s work on measure theory, Hurwitz’s and Hadamard’s work on number theory. Moreover, it was becoming obvious that set theory can unite mathe- matical disciplines like algebra, geometry, calculus into one, whilst heretofore they were developing largely along their own lines. Now many common themes and connections appeared, allowing new insights. Clearer foundations for set theory were needed. In 1904 David Hilbert, worried by the paradoxes, declared that Cantor left it too open for subjective judgement to decide which collections are sets and which are not sets. To eliminate this subjectivity he called for an axiomatization of the set theory. Various attempts were made to provide the , the most successful (in the sense of leading to the system mostly used today) due to Zermelo.

The original principle employed by the founders of set theory set theory was the Na¨ıveComprehension which asserted that each property P of objects x has an extension; i.e. a set of all the objects having the property - we have introduced the notation {x | x has P } for it. However, the paradoxes have shown that some properties cannot have extensions because such extensions would be contradictory, either through being too large or for some other reason. Zermelo developed his original axiom system (1908) by surveying what instances of Comprehension were actually used in mathematical practice. His informal system had comprehension axioms

Emptyset There is a set ∅ = {x | x 6= x}. Pairing For all objects a, b there is a set {a, b } = {x | x = a ∨ x = b}. Union For every set of sets a there is a set a = {x | x ∈ y for some y ∈ a}. S Powerset For every set a there is a set P ow (a) = {x | x ⊆ a}. Separation (Subset) For each definite property P of objects x and every set a there is a set {x ∈ a | x has P } = {x | x ∈ a &x has P }, the axiom of Infinity

Infinity There exists a set A such that ∅ ∈ A and whenever x ∈ A then {x} ∈ A. and an . Later the Replacement Axiom scheme suggested by Fraenkel was added

Replacement For every set a and each definite operation F assigning an object F (x) to each x ∈ a there is a set

{F (x) | x ∈ a} = {y | y = F (x) for some x ∈ a},

27 and the axiom system became understood in a more formal way by taking the definite properties to be those expressible in a suitable first order language. Today a somewhat different form of the axiom of Infinity is used, and another axiom, the Foundation axiom, is usually added to form the modern first or- der axiom system Zermelo-Fraenkel ( ZF ). This axiom system is intended to axiomatize a universe of pure, well-founded sets.

Some intuition about this universe can be gained from the picture below (which we will return to when we have a precise definition of ordinal numbers).

Cumulative Hierarchy (von Neuman’s sets)

. . . .

Vω+ω+1 = P ow (Vω+ω)

∞ Vω+ω = n=0 Vω+n S . . . . \ / \ / V ω+1 = P ow (Vω) \ / ∞ \ / V ω = n=0 Vn \ / S . . \ . / . \ / ∅,{∅} ,{∅ ,{∅}} ,{{∅}} \ / V 2 = P ow (V1) \ / \ ∅ , {∅} / V 1 = P ow (V0) \ / \ ∅ / V 0 \ / \ / \/

The pure sets We define a set to be pure if its elements are sets, their elements are sets and the elements of those elements are sets and so on; i.e. a set b is a

28 pure set if, whenever a1,...,a n are sets such that

an ∈ an−1 ∈ ··· ∈ a1 ∈ b then every element of an is a set.

The well-founded sets We define a set b to be ∈-descending if every element of b is a set that has an element that is also in b. A set a is well-founded if it is not an element of any ∈-descending set. The Foundation Axiom says that every set is well-founded, or equivalently, that there are no ∈-descending sets except - trivially - the empty set. In other words, any non-empty set b must have an element x ∈ b such that x ∩ b = ∅.

Observe that if a0, a 1,... is an infinite ∈-descending sequence of sets; that is,

· · · ∈ an+1 ∈ an ∈ ··· ∈ a1 ∈ a0, then the set {a0, a 1,... } is a non-empty ∈-descending set so that each an is a non-well-founded set. Also note that if b is a set such that b ∈ b then {b} is a non-empty ∈-descending set and so b is non-well-founded. More generally any circular set; i.e. any set b such that b ∈ an ∈ ··· ∈ a1 ∈ b, is a non-well-founded set. So the Foundation Axiom rules out all such sets. The intuition behind the Foundation Axiom is that sets should exist in some kind of conceptual order - a set can only exist if its elements pre-exist.

29