Topological forcing semantics with settling

Abstract

It was realized early on that topologies can model constructive systems, as the open sets form a Heyting algebra. After the development of forcing, in the form of Boolean-valued models, it became clear that, just as over ZF any Boolean-valued model also satisfies ZF, Heyting-valued models satisfy IZF, which stands for Intuitionistic ZF, the most direct constructive re-working of the ZF axioms. In this paper, we return to topologies, and introduce a variant model, along with a correspondingly revised forcing or satisfaction relation. The purpose is to prove independence results related to weakenings of the Power Set axiom. The original motivation is the second model of [9], based on R, which shows that Exponentiation, in the context of CZF minus Subset Collection, does not suffice to prove that the Dedekind reals form a set. The current semantics is the generalization of that model from R to an arbitrary topological space. It is investigated which set-theoretic principles hold in such models in general. In addition, natural properties of the underlying topological space are shown to imply the validity of stronger such principles.