Pseudo-countable models

  • J. D. Hamkins, “Pseudo-countable models,” mathematics arXiv, 2022.
    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},
    url = {},

Download pdf at arXiv:2210.04838

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.