On Transferring Model Theoretic Theorems of $${\mathcal{L}_{{\infty},\omega}}$$ L ∞ , ω in the Category of Sets to a Fixed Grothendieck Topos

Abstract

Working in a fixed Grothendieck topos Sh(C, JC) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, JC). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\), such as the downward Lowenheim–Skolem theorem, the completeness theorem and Barwise compactness.