By Vladimir Kanovei

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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. For brevity let us write w for w1 , . . , wn . For each formula Φ(v, w) of the language of set theory we have: Theorem 8. Φ For all w, if ∀n ∈ N [(∀m ∈ n Φ(m, w)) → Φ(n, w)] then ∀n ∈ N Φ(n, w).

This is possible since κ is a limit ordinal. It now suffices to prove that type <← { δ, δ }, < < κ. By Theorem 23 it suffices to prove that |<← { δ, δ }| < κ and so it suffices to prove that |δ × δ| < κ. Since |δ × δ| = ||δ| × |δ||, it suffices to prove that |λ × λ| < κ for all cardinals λ < κ. If λ is infinite, this is true by inductive hypothesis. If λ is finite, then |λ × λ| < ω ≤ κ. 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.

The fact that N ∈ ON now follows immediately from Theorem 6. , ω = N. Theorems 6 and 12 now show that the natural numbers are the smallest ordinals, which are immediately succeeded by ω, after which the rest follow. The other ordinals are generated by two processes illustrated by the next lemma. Lemma. 1. ∀α ∈ ON ∃β ∈ ON β = succ(α). 2. ∀S [S ⊆ ON → ∃β ∈ ON β = S]. Exercise 7. Prove this lemma. For S ⊆ ON we write sup S for the least element of {β ∈ ON : (∀α ∈ S)(α ≤ β)} if such an element exists.