 By W. Weiss

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Suppose that n is not a subset of m; using Foundation pick l ∈ n \ m such that l ∩ (n \ m) = ∅. By transitivity, l ⊆ n and hence l ⊆ m. Now by (1) applied to l and m, we conclude that l = m. Hence m ∈ n. These theorems show that “∈” behaves on N just like the usual ordering “<” on the natural numbers. In fact, we often use “<” for “∈” when writing about the natural numbers. We also use the relation symbols ≤, >, and ≥ in their usual sense. 34 CHAPTER 4. THE NATURAL NUMBERS The next theorem scheme justifies ordinary mathematical induction.

X3 − 1 = 3(3 3 +1) 3 + 33 + 33 . Change all 3’s to 4’s, leaving any 1’s or 2’s alone. x4 = 4(4 4 +1) 4 + 44 + 44 . Subtract 1. x4 − 1 = 4(4 4 +1) 4 + 44 + 3 · 43 + 3 · 42 + 3 · 4 + 3. Change all 4’s to 5’s, leaving any 1’s, 2’s or 3’s alone. x5 = 5(5 5 +1) 5 + 55 + 3 · 53 + 3 · 52 + 3 · 5 + 3. Subtract 1 and continue, changing 5’s to 6’s, subtracting 1, changing 6’s to 7’s and so on. One may ask the value of the limit lim xn . n→∞ What is your guess? The answer is surprising. Theorem 18. (Goodstein) For any initial choice of x there is some n such that xn = 0.

Suppose cf (λ) ≤ κ. Then λ = |P(κ)| = |κ 2| = |(κ×κ) 2| = |κ (κ 2)| = |κ λ| ≥ |cf (λ) λ| > λ. Cantor’s Theorem guarantees that for each ordinal α there is a set, P(α), which has cardinality greater than α. However, it does not imply, for example, that ω + = |P(ω)|. This statement is called the Continuum Hypothesis, and is equivalent to the third question in the introduction. 64 CHAPTER 7. CARDINALITY The aleph function ℵ : ON → ON is defined as follows: ℵ(0) = ω ℵ(α) = sup {ℵ(β)+ : β ∈ α}. We write ℵα for ℵ(α).