Realizability for Constructive Zermelo-Fraenkel Set Theory

Notice that e ⊩ u ∈ v and e ⊩ u = v can be defined for arbitrary sets u, v, viz., not just foru, v ∈ V(A). The definitions of e ⊩ u ∈ v and e ⊩ u = v fall under the scope of definitionsby transfinite recursion. More precisely, the functionsF ∈ (u, v) = {e ∈ |A| : e ⊩ u ∈ v}G = (u, v) = {e ∈ |A| : e ⊩ u = v}can be defined (simultaneously) on V × V by recursion on the relation〈c, d〉 ✁ 〈a, b〉 iff (c = a ∧ d ∈ TC(b)) ∨ (d = b ∧ c ∈ TC(a)).4.1 The soundness theorem for intuitionistic predicate logic with equalityExcept for the extra considerations concerning bounded quantifiers, the proofs of 4.2 and4.3 are almost the same for CZF as the corresponding proofs for IZF given in [16].Lemma: 4.2 There are i r , i s , i t , i 0 , i 1 ∈ |A| such that for all a, b, c ∈ V(A),1. i r ⊩ a = a.2. i s ⊩ a = b → b = a.3. i t ⊩ (a = b ∧ b = c) → a = c.4. i 0 ⊩ (a = b ∧ b ∈ c) → a ∈ c.5. i 1 ⊩ (a = b ∧ c ∈ a) → c ∈ b.Moreover, for each formula ϕ(v, u 1 , . . . , u r ) of CZF all of whose free variables are amongv, u 1 , . . . , u r there exists i ϕ ∈ |A| such that for all a, b, c 1 , . . . , c r ∈ V(A),where ⃗c = c 1 , . . . , c r .i ϕ ⊩ ϕ(a,⃗c) ∧ a = b → ϕ(b,⃗c),Proof: Realizers for the universal closures of the above formulas can be taken from [16],chapter 2, sections 5 and 6. Thus the above assertions follow from the “genericity” ofrealizers of universal statements, i.e.,e ⊩ ∀vψ(v) iff ∀a e ⊩ ψ(a).✷Theorem: 4.3 Let D be a proof in intuitionistic predicate logic with equality of a formulaϕ(u 1 , . . . , u r ) of CZF all of whose free variables are among u 1 , . . . , u r . Then there ise D ∈ |A| such that CZF provese D ⊩ ∀u 1 . . . ∀u r ϕ(u 1 , . . . , u r ).10