Sequential topologies and Dedekind finite sets

Topological Forcing Semantics with Settling

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, so do 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…