Definable Coherent Ultrapowers and Elementary Extensions

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 ${\mathbb{L}}_{\infty ,\ddot{I}‰}$ that defines Skolem functions by a sufficiently complete (but in $ZFC$) coherent ultrafilter. We apply this method to various elementary classes and AECs.