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But we will take the liberty of calling our constructs “the natural numbers”. We begin by taking 0 as the empty set ∅. We write 1 2 3 succ(x) for for for for {0} {0, 1} {0, 1, 2} x ∪ {x} We write “n is a natural number” for [n = ∅ ∨ (∃l ∈ n)(n = succ(l))] ∧ (∀m ∈ n)[m = ∅ ∨ (∃l ∈ n)(m = succ(l))] and write: N for {n : n is a natural number} The reader can gain some familiarity with these definitions by checking that succ(n) ∈ N for all n ∈ N. 31 32 CHAPTER 4. THE NATURAL NUMBERS We now begin to develop the basic properties of the natural numbers by introducing an important concept.

THE NATURAL NUMBERS Chapter 5 The Ordinal Numbers The natural number system can be extended to the system of ordinal numbers. An ordinal is a transitive set of transitive sets. More formally: for any term t, “t is an ordinal” is an abbreviation for (t is transitive) ∧ (∀x ∈ t)(x is transitive). We often use lower case Greek letters to denote ordinals. We denote {α : α is an ordinal} by ON. From Theorem 6 we see immediately that N ⊆ ON. Theorem 10. 1. ON is transitive. 2. ¬(∃z)(z = ON). Proof. 1.

Every set has a cardinality. Those ordinals which are |x| for some x are called cardinals. Exercise 19. Prove that each n ∈ ω is a cardinal and that ω is a cardinal. Show that ω + 1 is not a cardinal and that, in fact, each other cardinal is a limit ordinal. Theorem 23. The following are equivalent. 1. κ is a cardinal. 2. , |κ| = κ. 3. (∀α < κ)(¬∃ injection f : κ → α). 59 60 CHAPTER 7. CARDINALITY Proof. We prove the negations of each are equivalent: ¬(1) (∀x)[(∃ bijection g : x → κ) → (∃α < κ)(∃ bijection h : x → α)] ¬(2) ∃α < κ ∃ bijection f : κ → α ¬(3) ∃α < κ ∃ injection f : κ → α ¬(2) ⇒ ¬(1) Just take h = f ◦ g.

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