Zermelo-Fraenkel Set Theory

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CHAPTER 4. ORDINALS 32 1. Every V α is transitive, 2. x ⊂ y ∈ V α ⇒ x ∈ V α , 3. α < β ⇒ V α ∈ V β ; α β ⇒ V α ⊂ V β , 4. α ⊂ V α ; α ∉ V α ; α = OR ∩ V α , 5. OR ∩ (V α+1 − V α ) = {α}. The following results “explain” the Foundation Axiom. Theorem 4.18 G is the least fixed point of ℘ (and V is the greatest one). Proof. 1. ℘(G) ⊂ G. See Exercise 16 (p. 10). 2. ℘(X) ⊂ X ⇒ G ⊂ X: Aiming at a contradiction, assume that ℘(X) ⊂ X, a ∈ G, a ∉ X. Claim: TC(a) ⊂ X. (And it follows that a ⊂ X and a ∈ X: contradiction.) Proof: Assume that TC(a) − X ≠ ∅. Note that (since G is transitive: Exercise 16) a ⊂ G and hence TC(a) ⊂ G (again Exercise 16). Thus, if some b is ∈ TC(a) − X, it is in G as well; and therefore some b ∈ TC(a) − X has b ∩ (TC(a) − X) = ∅. But then b ⊂ X and b ∈ X. □ Corollary 4.19 The following are equivalent: 1. The Foundation Axiom, 2. ⋃ α V α = V, 3. every non-empty class is disjoint with some of its elements, 4. (∈-Induction) for every class K, if ∀a [∀b(b ∈ a ⇒ b ∈ K) ⇒ a ∈ K], then V ⊂ K. Proof. 1 ⇔ 2. Immediate, since Foundation says that V = G, Theorem 4.18 says that G = ℘↑, and Theorem 4.15.2 implies that ℘↑= ⋃ α V α. 1 ⇒ 4. Note that ∈-induction is nothing but the implication ℘(K) ⊂ K ⇒ V ⊂ K. By Theorem 4.18, this holds when V is replaced by G. However, Foundation says that V = G. 3 ⇔ 4. These principles are logically equivalent (take complements of the classes involved). 3 ⇒ 1. Trivial, since every set is a class. □ Definition 4.20 For a ∈ ⋃ α V α, define the rank ρ(a) of a by: ρ(a) := ⋂ {α | a ⊂ V α }. □ If Foundation is assumed, you can define an operation on non-empty classes that uniformly selects a non-empty subset. Definition 4.21 Bottom(X) := {x ∈ X | ∀y ∈ X(ρ(x) ρ(y))}. Lemma 4.22 1. Bottom(X) ⊂ X, 2. Bottom(X) ∈ V, 3. if X ≠ ∅, then Bottom(X) ≠ ∅. □
CHAPTER 4. ORDINALS 33 Exercises 73 ♣ Show that V α = ⋃ ξ
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Page 4 and 5: CONTENTS 2 7.6 Consistency of V = L
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CHAPTER 8. FORCING 82 Note that for
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CHAPTER 8. FORCING 84 Some examples
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CHAPTER 8. FORCING 86 Definition 8.
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CHAPTER 8. FORCING 88 Exercises 226
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CHAPTER 8. FORCING 90 supposed exis
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CHAPTER 8. FORCING 92 Lemma 8.37 M[
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CHAPTER 8. FORCING 94 8.3 Alternati
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CHAPTER 8. FORCING 96 Proof of Lemm
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CHAPTER 8. FORCING 98 8.4 Faits div
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CHAPTER 8. FORCING 100 8.4.3 A few
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CHAPTER 8. FORCING 102 Thus, ccc eq
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CHAPTER 8. FORCING 104 246 ♣ Exer
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CHAPTER 8. FORCING 106 Note that MA
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CHAPTER 8. FORCING 108 Proof. Choos
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CHAPTER 8. FORCING 110 Therefore, G
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Solutions to Exercises Chapter 2 5.
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2. If x ∈ y ∈ TC(a), then, for
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1. Her(B) is the greatest (post-) f
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(ii) if H + (K) ⊂ K and ∀ξ <
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119. (Bukovsky; Hechler) Assume tha
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B by x ≺ y ≡ the
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Zermelo-Fraenkel Set Theory

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