- J. D. Hamkins, “Pseudo-countable models,” mathematics arXiv, 2022.

[Bibtex]`@ARTICLE{Hamkins:Pseudo-countable-models, author = {Joel David Hamkins}, title = {Pseudo-countable models}, journal = {mathematics arXiv}, year = {2022}, volume = {}, number = {}, pages = {}, month = {}, note = {manuscript under review}, abstract = {}, keywords = {under-review}, source = {}, doi = {}, eprint = {2210.04838}, archivePrefix={arXiv}, primaryClass={math.LO}, url = {http://jdh.hamkins.org/pseudo-countable-models}, }`

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**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.