Abstract
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
$\mathsf {IZF}$
.
Publisher
Cambridge University Press (CUP)