Realizability for Constructive Zermelo-Fraenkel Set Theory

From (32) it follows that u ∈ A, and thus 〈0, u〉 ∈ C. So we getLetting s(f) := λe.pe( ˜fe) 1 , (31) and (32) yieldp0i r ⊩ u ∈ C. (33)s(f) ⊩ ∀x ∈ d ∃y ∈ u ϕ(x, y),s(f) ⊩ ∀y ∈ u ∃x ∈ d ϕ(x, y).By invoking Lemma 4.2, from the latter we can effectively determine application terms˜s(f) and ŝ(f) such that˜s(f) ⊩ ∀x ∈ b ∃y ∈ u ϕ(x, y), (34)ŝ(f) ⊩ ∀y ∈ u ∃x ∈ b ϕ(x, y). (35)Hence, letting ˜q := λf.p(p0i r )(p˜s(f)ŝ(f)), (33), (34), and (35) entail that˜q ⊩ ∀b(b ∈ C ∧ ∀x ∈ b ∃y ∈ C ϕ(x, y) → (36)∃u ∈ C [∀x ∈ b ∃y ∈ u ϕ(x, y) ∧ ∀y ∈ u ∃x ∈ b ϕ(x, y)]).Note that the entire construction of ˜q shows that it neither depends on C nor on theparameters of ϕ. Choosing ϕ(x, y) to be the formula r ⊆ b × C ∧ 〈x, y〉 ∈ r, we deducefrom (36) and (24) thatThus, in view of (23), we getp(p ˜mñ)˜q ⊩ Reg(C).p(p0i r )(p(p ˜mñ)˜q) ⊩ ∀a ∃C [a ∈ C ∧ Reg(C)].✷7 Adding Markov’s Principle and Independence of PremissesMarkov’s Principle (MP) is closely associated with the work of the school of Russianconstructivists. The version of MP most appropriate to the set-theoretic context is theschema∀n ∈ ω [ϕ(n) ∨ ¬ϕ(n)] → [¬¬∃n ∈ ω ϕ(n) → ∃n ∈ ωϕ(n)].The variant¬¬∃n ∈ ω R(n) → ∃n ∈ ωR(n),with R being a primitive recursive predicate, will be denoted by MP PR .MP PR is implied by MP.Obviously,20