By Michael F. Barnsley

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It follows that the Baire space is homeomorphic to the space of all irrational numbers. For various reasons, modern descriptive set theory uses the Baire space rather than the real line. Often the functions in ω ω are called reals. 8 above. The Cantor-Bendixson Theorem holds as well. For completeness we give a description of perfect sets in N . 4. Real Numbers 43 Let Seq denote the set of all ﬁnite sequences of natural numbers. 4) if t ∈ T and s = t n for some n, then s ∈ T . 5) [T ] = {f ∈ N : f n ∈ T for all n ∈ N }.

11. 26 (Cantor’s Normal Form Theorem). Every ordinal α > 0 can be represented uniquely in the form α = ω β1 · k1 + . . + ω βn · kn , where n ≥ 1, α ≥ β1 > . . > βn , and k1 , . . , kn are nonzero natural numbers. Proof. By induction on α. For α = 1 we have 1 = ω 0 · 1; for arbitrary α > 0 let β be the greatest ordinal such that ω β ≤ α. 25(iv) there exists a unique δ and a unique ρ < ω β such that α = ω β · δ + ρ; this δ must necessarily be ﬁnite. The uniqueness of the normal form is proved by induction.

1. The relation “(P, <) is isomorphic to (Q, <)” is an equivalence relation (on the class of all partially ordered sets). 2. α is a limit ordinal if and only if β < α implies β + 1 < α, for every β. 3. If a set X is inductive, then X ∩ Ord is inductive. The set N = {X : X is inductive} is the least limit ordinal = 0. 26 Part I. 4. (Without the Axiom of Inﬁnity). Let ω = least limit α = 0 if it exists, ω = Ord otherwise. Prove that the following statements are equivalent: (i) There exists an inductive set.