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This is illustrated by the following examples, which are easy to verify from the basic definitions. Examples. 1. 1 + ω = 2 + ω 2. 1 + ω = ω + 1 3. 1 · ω = 2 · ω 4. 2 · ω = ω · 2 5. 2ω = 4ω 6. (2 · 2)ω = 2ω · 2ω Lemma. If β is a non-zero ordinal then ω β is a limit ordinal. Exercise 11. Prove this lemma. Lemma. If α is a non-zero ordinal, then there is a largest ordinal β such that ω β ≤ α. Exercise 12. Prove this lemma. Show that the β ≤ α and that there are cases in which β = α. ) Lemma. γ ∈ ON α = β + γ.

Exercise 16. Prove that a transitive set α is an ordinal iff the membership relation is total on α. Suppose δ is an ordinal and f : X → δ. A relation R on X with the property that f (x) < f (y) whenever x, y ∈ R must be a well founded relation. In fact, this turns out to be a characterisation. Theorem 19. Let R be a relation on a set X. R is well founded iff there is an ordinal δ and a surjection f : X → δ such that f (x) < f (y) whenever x, y ∈ R. Proof. We treat only the forward implication. Using recursion on ON we define g : ON → P(X) by g(β) = x : x is a minimal element of X \ {g(α) : α < β} .

Since well orders are extensional and well founded we can use the Mostowski Collapsing Theorem. By Exercise 16 the resulting transitive set is an ordinal. The unique ordinal given by this theorem is called the order type of the well ordered set. We denote the order type of X, < by type X, < . 57 We now come to the Well Ordering Principle, which is the fundamental theorem of Set Theory due to E. Zermelo. In order to prove it we use the Axiom of Choice and, for the first time, the Power Set Axiom.

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An Introduction to Set Theory by W. Weiss


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