Abstract
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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Ackerman, N.L. On Transferring Model Theoretic Theorems of \({\mathcal{L}_{{\infty},\omega}}\) in the Category of Sets to a Fixed Grothendieck Topos. Log. Univers. 8, 345–391 (2014). https://doi.org/10.1007/s11787-014-0105-5
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DOI: https://doi.org/10.1007/s11787-014-0105-5