Realizability for Constructive Zermelo-Fraenkel Set Theory

consequence, there is an index d ∗ calculable from d such that d ∗ ⊩ x ∈ B ∗ . Therefore wehave all the ingredients to compose ê as claimed in (50).To show realizable functionality of ˜F , suppose f ⊩ 〈b, c〉 Kl ∈ ˜F and h ⊩ 〈b, d〉 Kl ∈ ˜F .Then there exist 〈n, x〉, 〈m, y〉 ∈ B ∗ such that ((f) 0 ) 1 = n, ((h) 0 ) 1 = m, and(f) 1 ⊩ 〈b, c〉 Kl = 〈x, F (n, x)〉 Kl ∧ (h) 1 ⊩ 〈b, d〉 Kl = 〈y, F (m, y)〉 Kl . (52)From (52) one gets V(Kl) |= x = y. Moreover, as 〈n, x〉, 〈m, y〉 ∈ B ∗ there are u, v ∈ Bsuch thatx = 〈j 0 (u), u st 〉 Kl and n = j 0 (u), andy = 〈j 0 (v), v st 〉 Kl and m = j 0 (v).Since V(Kl) |= x = y, the foregoing yields V(Kl) |= n = m ∧ u st = v st , and so, byProposition 8.4 and (44), we can conclude that n = m and u = v, and hence x = y andF (n, x) = F (m, y). Thus, in view of (52), there is a partial recursive function ν such thatν(f, h) ⊩ c = d, verifying functionality of ˜F , i.e., V(Kl) |= ˜F is a function.Combining the latter result with (50) and (49) allows one to construct the desired e ′from e such that (48) holds.✷11 Continuity PrinciplesFundamental to Brouwer’s development of intuitionistic mathematics are strong continuityprinciples incompatible with classical mathematics.Definition: 11.1 Some continuity principles which pertain to Brouwer’s mathematicsare: 11. Cont(N N , N): Every function from N N to N is continuous.2. ∀X∀Y Cont(X, Y): For every complete separable metric space X and separable metricspace Y, Cont(X, Y), i.e., every function from X to Y is continuous.Recall that MP PR is Markov’s principle for primitive recursive predicates.Theorem: 11.2 (CZF + MP PR ). V(Kl) |= Cont(N N , N) ∧ ∀X∀Y Cont(X, Y).Moreover, V(Kl) validates that every separable metric space is subcountable.Proof: For the proof that MP is realized in V(Kl) it suffices to have MP PR in thebackground universe. As a result of this and Theorem 9.2, we have V(Kl) |= MP ∧ ECT.By [7] IV.3.1, MP PR proves KLS, where KLS stands for the Kreisel-Lacombe-Shoenfield’s theorem asserting that every effective operation from N N to N is continuous.By [7] XVI.2.1.1, ECT implies KLS → Cont(N N , N). In consequence, V(Kl) |=Cont(N N , N).1 For exact formalizations of the notions of complete metric space, separable metric space, and continuityin constructive set theory see [7], chap. I, section 20.32