Preservation of choice principles under realizability

Especially nice models of intuitionistic set theories are realizability models V(A)V(A)⁠, where AA is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in V(A)V(A) if assumed in the underlying universe VV⁠, adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axioms holds in V(A)V(A) for every applicative structure AA if it holds in the background universe.1 It is also shown that a weak form of the countable axiom of choice, ACω,ωACω,ω⁠, is rendered true in any V(A)V(A) regardless of whether it holds in the background universe. The paper extends work by McCarty (1984, Realizability and Recursive Mathematics, PhD Thesis) and Rathjen (2006, Realizability for constructive Zermelo–Fraenkel set theory.