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

Logica Universalis 8 (3-4):345-391 (2014)
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Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). 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 Löwenheim–Skolem theorem, the completeness theorem and Barwise compactness



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Relativized Grothendieck topoi.Nathanael Leedom Ackerman - 2010 - Annals of Pure and Applied Logic 161 (10):1299-1312.

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