Realizability for Constructive Zermelo-Fraenkel Set Theory

The class Y is Φ-closed if Γ Φ(Y ) ⊆ Y . Note that Γ Φis monotone; i.e. for classes Y 1 , Y 2 ,whenever Y 1 ⊆ Y 2 , then Γ Φ(Y 1 ) ⊆ Γ Φ(Y 2 ).We define the class inductively defined by Φ to be the smallest Φ-closed class. Themain result about inductively defined classes states that this class, denoted I(Φ), alwaysexists.Lemma: 3.3 (CZF) (Class Inductive Definition Theorem) For any inductive definitionΦ there is a smallest Φ-closed class I(Φ).Moreover, there is a class J ⊆ ON × V such thatI(Φ) = ⋃ αJ α ,and for each α,J α = Γ Φ( ⋃ β∈αJ β ).J is uniquely determined by the above, and its stages J α will be denoted by Γ α Φ .Proof: [2], section 4.2 or [4], Theorem 5.1.✷Lemma: 3.4 The classes V(A) α are definable in CZF.Proof: Let Φ be the inductive definition withxa Φ iff ∀u∈a (u ∈ |A| × x).Invoking Lemma 3.3, let J be the class such that I(Φ) = ⋃ α J α , and for each α,J α = Γ Φ( ⋃ β∈αJ β ).Now let∆ α := ⋃ β∈αJ β .Note that Γ Φ(X) = P(|A| × X), and therefore∆ α = ⋃ β∈αJ β (3)= ⋃ Γ Φ( ⋃ J η )β∈α η∈β= ⋃ P(|A| × ⋃ J η )β∈αη∈β= ⋃ β∈αP(|A| × ∆ β ).Letting V(A) α := ∆ α , (3) shows that the equations of definition 3.1 obtain. ✷8