A Reduction Theorem for the Kripke–Joyal Semantics: Forcing Over an Arbitrary Category can Always be Replaced by Forcing Over a Complete Heyting Algebra

Abstract

It is assumed that a Kripke–Joyal semantics ${\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}$ has been defined for a first-order language ${\mathcal{L}}$. To transform ${\mathbb{C}}$ into a Heyting algebra ${\overline{\mathbb{C}}}$ on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of ${\mathbb{C}}$. A pretopology ${\overline{{\rm Cov}}}$ is defined on ${\overline{\mathbb{C}}}$ using the pretopology on ${\mathbb{C}}$. A sheaf ${\overline{{\it F}}}$ is made up of sections of F that obey functoriality. A forcing relation ${\overline{\Vdash}}$ is defined and it is shown that ${\overline{\mathcal{A}} = \left\langle \overline{\mathbb{C}},\overline{\rm{Cov}},\overline{{\it F}}, \overline{\Vdash} \right\rangle }$ is a Kripke–Joyal semantics that faithfully preserves the notion of forcing of ${\mathcal{A}}$. That is to say, an object a of ${\mathbb{C}Ob}$ forces a sentence with respect to ${\mathcal{A}}$ if and only if the maximal a-crible forces it with respect to ${\overline{\mathcal{A}}}$. This reduces a Kripke–Joyal semantics defined over an arbitrary site to a Kripke–Joyal semantics defined over a site which is based on a complete Heyting algebra.