By W. Weiss

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Booklet by way of Sigler, L. E.

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There is an implicit formula for the calculation of y = F (x) which is [x = 0 → y = 3] ∧ (∀n ∈ N)[x = succ(n) → y = succ(F (n))] However the formula involves F , the very thing that we are trying to describe. Is this a vicious circle? No — the formula only involves the value of F at a number n less than x, not F (x) itself. In fact, you might say that the formula doesn’t really involve F at all; it just involves F |x. Let’s rewrite the formula as [x = 0 → y = 3] ∧ (∀n)[x = succ(n) → y = succ(f (n))] and denote it by Φ(x, f, y).

62 CHAPTER 7. CARDINALITY Corollary. If X is infinite, then |X × Y | = max {|X|, |Y |}. Theorem 26. For any X we have | X| ≤ max {|X|, sup {|a| : a ∈ X}}, provided that at least one element of X ∪ {X} is infinite. Proof. Let κ = sup {|a| : a ∈ X}. Using Exercise 20, for each a ∈ X there is a surjection fa : κ → a. Define a surjection f : X × κ → X by f ( a, α ) = fa (α). Using Exercise 20 and the previous corollary, the result follows. Define B A as {f : f : B → A} and [A]κ as {x : x ⊆ A ∧ |x| = κ}.

2 · 2)ω = 2ω · 2ω Lemma. If β is a non-zero ordinal then ω β is a limit ordinal. Exercise 11. Prove this lemma. Lemma. If α is a non-zero ordinal, then there is a largest ordinal β such that ω β ≤ α. Exercise 12. Prove this lemma. Show that the β ≤ α and that there are cases in which β = α. ) Lemma. γ ∈ ON α = β + γ. Exercise 13. Prove this lemma. Commonly, any function f with dom(f ) ⊆ ω is called a sequence. If dom(f ) ⊆ n + 1 for some n ∈ ω, we say that f is a finite sequence; otherwise f is an infinite sequence.

### An Introduction to Set Theory by W. Weiss

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