Between reduced powers and ultrapowers, II

TL;DR

This paper demonstrates, within ZFC, that ultraproducts of certain infinite structures are not isomorphic to reduced products associated with the Fréchet filter, highlighting limitations of ultraproducts in model theory.

Contribution

It establishes the consistency of non-isomorphism between ultraproducts and reduced products for specific structures, extending understanding of their differences.

Findings

Ultraproducts of countably infinite structures are not isomorphic to certain reduced products.

Countably saturated structures are isomorphic under CH when elementarily equivalent.

The results depend on set-theoretic assumptions like ZFC and CH.

Abstract

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fr\'echet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.