By W. Weiss

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**Introduction to Set Theory (International Series in Pure and Applied Mathematics)**

This ebook is inteded to be a self-contained advent to all of the set idea wanted by way of such a lot mathematicians. The method of set conception here's axiomatic. Logical symbolism is used, yet merely the place it truly is crucial, or the place it kind of feels to elucidate a scenario. Set idea can be in keeping with formal good judgment, yet right here it truly is in response to intuitive common sense.

Set thought has skilled a quick improvement in recent times, with significant advances in forcing, internal types, huge cardinals and descriptive set thought. the current ebook covers every one of those parts, giving the reader an realizing of the guidelines concerned. it may be used for introductory scholars and is large and deep sufficient to carry the reader close to the limits of present examine.

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**Example text**

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”.