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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. Prove this lemma. We can perform induction on the ordinals via a process called transfinite induction.

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 13. Prove this lemma. Commonly, any function f with dom(f ) ⊆ ω is called a sequence.

Exercise 20. Prove the following: 1. |x| = |y| iff ∃ bijection f : x → y. 2. |x| ≤ |y| iff ∃ injection f : x → y. 3. |x| ≥ |y| iff ∃ surjection f : x → y. Assume here that y = ∅. Theorem 24. (G. Cantor) ∀x |x| < |P(x)|. Proof. First note that if |x| ≥ |P(x)|, then there would be a surjection g : x → P(x). But this cannot happen, since {a ∈ x : a ∈ / g(a)} ∈ / g → (x). 61 For any ordinal α, we denote by α+ the least cardinal greater than α. This is well defined by Theorem 24. Exercise 21. Prove the following: 1.