Abstract In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$ . We then show that adding such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$ .

Abstract In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe $\mathrm{V_{ex}}(A)$ in which the axiom of choice in all finite types, ${\textsf{AC}}_{{\textsf{FT}}}$, is realized, where $A$ stands for an arbitrary partial combinatory algebra. This construction furnishes ‘inner models’ of many set theories that additionally validate ${\textsf{AC}}_{{\textsf{FT}}}$, in particular it provides a self-validating semantics for ${\textsf{CZF}}$ (constructive Zermelo–Fraenkel set theory) and ${\textsf{IZF}}$ (intuitionistic Zermelo–Fraenkel set theory). One can also add large set axioms and many other principles.

In this paper we continue the study, from Frittaion, Steila and Yokoyama (2017, Theory and Applications of Models of Computation 14th Annual Conference, Bern, Switzerland, April 20–22, 2017), on size-change termination (SCT) in the context of Reverse Mathematics. We analyse the soundness of the SCT method. In particular, we prove that the statement ‘any programme which satisfies the combinatorial condition provided by the SCT criterion is terminating’ is equivalent to WO(ω3) over RCA0.