  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.
