Pseudo-countable models

TL;DR

This paper introduces pseudo-countable models, which are uncountable structures viewed as countable within certain set-theoretic models, enabling the extension of countable model results to uncountable models.

Contribution

It proves that every pseudo-countable model admits extensions satisfying key set-theoretic theorems, broadening the scope of model-theoretic results to uncountable structures.

Findings

Pseudo-countable models can be extended to models of ZFC+V=L.

The class of pseudo-countable models is closed under forcing and interpretations.

Key theorems like Barwise extension and Keisler-Morley hold in pseudo-countable models.

Abstract

Every mathematical structure has an elementary extension to a pseudo-countable structure, one that is seen as countable inside a suitable class model of set theory, even though it may actually be uncountable. This observation, proved easily with the Boolean ultrapower theorem, enables a sweeping generalization of results concerning countable models to a rich realm of uncountable models. The Barwise extension theorem, for example, holds amongst the pseudo-countable models -- every pseudo-countable model of ZF admits an end extension to a model of ZFC+V=L. Indeed, the class of pseudo-countable models is a rich multiverse of set-theoretic worlds, containing elementary extensions of any given model of set theory and closed under forcing extensions and interpreted models, while simultaneously fulfilling the Barwise extension theorem, the Keisler-Morley theorem, the resurrection theorem, and the universal finite sequence theorem, among others.