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Ordinals. Transfinite Induction and Recursion We Will Not Use The 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.
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