Lifschitz Realizability as a Topological Construction

Abstract

We develop a number of variants of Lifschitz realizability for $\mathbf {CZF}$ by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of the lesser limited principle of omniscience including consistency with unique Church’s thesis, consistency with some Brouwerian principles and variants of the numerical existence property.

Rethinking the notion of oracle

“…By the correspondence theorem, this idea is absorbed by the notion of realizability relative to Lawvere-Tierney topologies. The reason why realizability relative to a Lawvere-Tierney topology has nice properties seems that a Lawvere-Tierney topology yields a subtopos, whose internal logic has a relationship with realizability relative to the topology; see also [20,29].…”