Set Theory (MATH 6730) Ordinals. Transfinite Induction and Recursion We will not use the Axiom of Choice in this chapter. All theorems we prove will be theorems of ZF = ZFC n fACg. 1. Ordinals To motivate ordinals, recall from elementary (naive) set theory that natural numbers can be used to label elements of a finite set to indicate a linear ordering: 0th, 1st, 2nd, :::, 2019th element. Although this will not be immediately clear from the definition, `ordinal numbers' or `ordi- nals' are sets which are introduced to play the same role for special linear orderings, called `well-orderings', of arbitrary sets. For example, one such ordering might be the following: 0th, 1st, 2nd, :::, 2019th, :::;!th, (! + 1)st element: | {z } we have run out of all natural numbers Definition 1.1. A set s is called transitive, if every element of s is a subset of s; i.e., for any sets x; y such that x 2 y 2 s we have that x 2 s. Thus, the class of all transitive sets is defined by the formula tr(s) ≡ 8x 8y (x 2 y ^ y 2 s) ! x 2 s: Definition 1.2. A transitive set of transitive sets is called an ordinal number or ordinal. Thus, the class On of all ordinals is defined by the formula on(x) ≡ tr(x) ^ 8y 2 x tr(y): Ordinals will usually be denoted by the first few letters of the Greek alphabet. The following facts are easy consequences of the definition of ordinals. 1 2 Theorem 1.3. (i) ; 2 On. (ii) If α 2 On, then α [ fαg 2 On. (iii) If α 2 On then α ⊆ On; i.e., every member of an ordinal is an ordinal. (iv) If A is a set of ordinals, then S A 2 On. (v) If A is a nonempty set of ordinals, then T A 2 On. Theorem 1.3(iii) shows that if On was a set, it would be an ordinal, and hence we would get On 2 On, which is impossible. This proves: Corollary 1.4. On is not a set. Definition 1.5. Let A be an ordinal or On, and define the class relations <A and ≤A on A by the formulas x <A y ≡ x 2 A ^ y 2 A ^ x 2 y and x ≤A y ≡ x 2 A ^ y 2 A ^ (x 2 y _ x = y): The subscript A will be omitted if A is clear from the context. 3 Theorem 1.6. (i) If A is an ordinal or A = On, then < is a (strict) linear order on A; that is, ZF ` 8x 2 A : x < x ^ 8x 2 A 8y 2 A 8z 2 A (x < y ^ y < z) ! x < z | {z } | {z } < is irreflexive < is transitive ^ 8x 2 A 8y 2 A (x = y _ x < y _ y < x) : | {z } the trichotomy law holds for < (ii) For arbitrary ordinals α and β we have that • α ≤ β if and only if α ⊆ β, and • α < β if and only if α ( β. (iii) For any set A of ordinals, S A is the least upper bound for A with respect to ≤; that is, the ordinal S A satisfies • α ≤ S A for all α 2 A, and • for every ordinal γ such that α ≤ γ for all α 2 A, we have that S A ≤ γ. (iv) For any nonempty set A of ordinals, T A is the least element of A with respect to ≤; that is, the ordinal T A satisfies • T A 2 A, and • T A ≤ α for all α 2 A. (v) There do not exist ordinals α; β such that α < β < α [ fαg. (vi) For arbitrary ordinals α and β we have that • α < β if and only if α [ fαg ≤ β. Idea of Proof. All statements in (i){(vi), except the trichotomy of < in (i), are fairly straight- forward consequences of the definition of ordinals and properties of ordinals proved in The- orem 1.3 or earlier items of this theorem. 4 We prove the trichotomy law for < on A by contradiction. So, assume that the trichotomy law fails for < on A, and choose α; β 2 A such that τ(α; β), where τ(x; y) is the formula τ(x; y) ≡ : x = y ^ : x 2 y ^ : y 2 x: Prove the following claims: 1. The set A = (α [ fαg) [ (β [ fβg) is an ordinal. 2. The set C = fx 2 A : 9y 2 A τ(x; y)g has an element γ such that γ \ C = ;. Fix such a γ. 3. The set D = fy 2 A : τ(γ; y)g has an element δ such that δ \ D = ;. Fix such a δ, and note that we have τ(γ; δ) by construction, so (∗) γ 6= δ; γ2 = δ; and δ2 = γ: 4. Now prove γ = δ by arguing that every element of γ is an element of δ, and every element of δ is an element of γ. We have reached the desired contradiction, which proves the trichotomy of < on A. 5 The statement in Theorem 1.6(v) motivates the following terminology: for every ordinal α, we call the ordinal α [ fαg the successor of α, and denote it by α +0 1.1 By combining statements (i) and (iv) of Theorem 1.6 we get that for every ordinal α, the relation < is a linear order on α such that every nonempty subset of α has a least element. Such a linear order is called a well-order. Since ; ⊆ x for every set, we see from Theorem 1.6(ii) that ; is the least ordinal; therefore we will also denote it by 0 (zero). The successor of 0 is the set ; [ f;g = f;g, which will be denoted by 1, and the successor of 1, which is the set f;g [ ff;gg = f;; f;gg, will be denoted by 2, etc. Definition 1.7. Let α 2 On. We will call α a successor ordinal if α = β +0 1 = β [ fβg for some ordinal β, and we will call α a limit ordinal if α 6= 0 and α is not a successor ordinal. Theorem 1.8. Let α 2 On. (i) The following conditions on α are equivalent: (a) α is a limit ordinal; (b) α 6= 0 and for every ordinal β < α there exists an ordinal γ such that β < γ < α; (c)0 6= α = S α. (ii) If α is a successor ordinal and α = β +0 1, then S α = β. 1After introducing addition for ordinals, the ordinal α +0 1 = α [ fαg will turn out to be the sum of α and 1 where 1 is the successor of 0 def= ;. 6 By construction, 1, 2, etc. are successor ordinals. Now we will use the Axiom of Infinity to show that there exists a limit ordinal. Let us call a set u inductive if it is a member of the class defined by the (abbreviated) formula ι(u) ≡ ; 2 u ^ 8x 2 u x [ fxg 2 u: The Axiom of Infinity is the statement that there exists an inductive set u. Let \ ! = fv 2 P(u): v is inductiveg = fx 2 u : 8v (ι(v) ! x 2 v)g: To justify the second equality notice that ⊇ is clear, while ⊆ holds, because if w is an inductive set, so is w \ u 2 P(u). Thus, • ! is a set (by Comprehension); • ! is inductive (as the intersection of any nonempty set of inductive sets is inductive); and • ! is a subset of all inductive sets. This shows that ! is the least inductive set with respect to ⊆; hence it is independent of the choice of u. Definition 1.9. The elements of ! are called natural numbers, so ! is the set of all natural numbers. The fact that ! has no proper subset that is inductive, is the Induction Principle for !: Theorem 1.10. For every subset A of !, if • 0 2 A, and • whenever y 2 A then y +0 1 = y [ fyg 2 A, then A = !. We can use induction on ! to prove the following properties of !: Theorem 1.11. ! is a limit ordinal. In fact, ! is the smallest limit ordinal with respect to <; that is, every ordinal β < ! is either 0 or a successor ordinal. 7 2. Well-founded and Set-like Class Relations Our goal is to extend • the Induction Principle for !, and • the familiar idea of constructing infinite sequences (i.e., functions with domain !) by recursion to all ordinals. It turns out that induction and recursion work | and are useful | in a much broader context, which we will discuss now. Let R be a class relation, and let A be a class; let R be defined by the formula '(x; y), and let A be defined by the formula (x). Definition 2.1. We say that R is well-founded on A if R ⊆ A × A and for every nonempty subset X of A there exists an element x 2 X such that there is no y 2 X satisfying (y; x) 2 R. Such an element x 2 X is called an R-minimal element of X. (It is useful to think of (u; v) 2 R as saying that \u R-precedes v" or \u is R-smaller than v", although R need not be an order or partial order.) There is a formula saying that \R is well-founded on A", namely: (1) 8x 8y '(x; y) ! ( (x) ^ (y)) ^ 8X (X 6= ; ^ 8x 2 X (x)) ! 9x 2 X 8y 2 X :'(y; x): Example 2.2.