An Introduction to Set Theory by W. Weiss

By W. Weiss

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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. Lemma. ∀S [S ⊆ ON → S = sup S] Exercise 8. Prove this lemma. An ordinal α is called a successor ordinal whenever ∃β ∈ ON α = succ(β). If α = sup α, then α is called a limit ordinal. 44 CHAPTER 5. THE ORDINAL NUMBERS Lemma. Each ordinal is either a successor ordinal or a limit ordinal, but not both. Exercise 9.

All natural numbers are equal. Proof. It is sufficient to show by induction on n ∈ N that if a ∈ N and b ∈ N and max (a, b) = n, then a = b. If n = 0 then a = 0 = b. Assume the inductive hypothesis for n and let a ∈ N and b ∈ N be such that max (a, b) = n + 1. Then max (a − 1, b − 1) = n and so a − 1 = b − 1 and consequently a = b. 40 CHAPTER 4. 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.

30 CHAPTER 3. THE AXIOMS OF SET THEORY Chapter 4 The Natural Numbers We now construct the natural numbers. That is, we will represent the natural numbers in our universe of set theory. We will construct a number system which behaves mathematically exactly like the natural numbers, with exactly the same arithmetic and order properties. We will not claim that what we construct are the actual natural numbers—whatever they are made of. But we will take the liberty of calling our constructs “the natural numbers”.

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