Definable Coherent Ultrapowers and Elementary Extensions

TL;DR

This paper introduces coherent ultrafilters and ultraproducts to characterize model extensions in fragments of logic, broadening the understanding of elementary extensions and applications to AECs.

Contribution

It develops the concept of coherent ultrafilters and uses definable ultraproducts to characterize model extensions in various logical fragments.

Findings

Characterization of model extensions via definable coherent ultraproducts.

Application of the method to elementary classes and AECs.

Framework without normality or well-foundedness in ultrafilters.

Abstract

We develop the notion of coherent ultrafilters (extenders without normality or well-foundedness). We then use definable coherent ultraproducts to characterize any extension of a model $M$ in any fragment of $\mathbb{L}_{\infty, \omega}$ that defines Skolem functions by a sufficiently complete (but in $ZFC$) coherent ultrafilter. We apply this method to various elementary classes and AECs.