Ultraproducts in abstract categorical logic

TL;DR

This paper extends the framework of abstract categorical logic by generalizing the ultraproduct method and proving Los's theorem within this setting, independent of specific quantifiers.

Contribution

It introduces a generalized ultraproduct construction and proves Los's theorem in the context of abstract categorical logic, broadening classical model theory methods.

Findings

Generalization of ultraproducts in categorical logic

Proof of Los's theorem in an abstract setting

Independence of ultraproducts from specific quantifiers

Abstract

In a previous publication, we introduced an abstract logic via an abstract notion of quantifier. Drawing upon concepts from categorical logic, this abstract logic interprets formulas from context as subobjects in a specific category, e.g., Cartesian, regular, or coherent categories, Grothendieck, or elementary toposes. We proposed an entailment system formulated as a sequent calculus which we proved complete. Building on this foundation, our current work explores model theory within abstract logic. More precisely, we generalize one of the most important and powerful classical model theory methods, namely the ultraproduct method, and show its fundamental theorem, i.e., Los's theorem. The result is shown as independently as possible of a given quantifier.