# How do we define countability of models?

I looked through all the other questions about countable models but none of them seem to really answer my question. I understand countability as a property from within a model - i.e. A set is countable if there is a bijection in that model from the set to the naturals. However, how do we define countability of a model?

A model of anything is a set, together with some extra stuff, with some properties - e.g. a model of ZFC is a set $M$ together with a binary relation $E\subseteq M^2$ satisfying the ZFC axioms.
• @exfret Well, this is ultimately going to depend on your foundational approach to things. If you're a Platonist, and believe that there is a "true" universe of sets $V$, then "countable" just means "countable in $V$." There are other approaches, but this is the simplest one (albeit the one most philosophically loaded and suspect). Sep 11, 2017 at 22:05
• @exfret Yes, I would argue that it is. It is, for example, a corollary of certain multiverse views. Or at least, a version of the statement "all sets are countable" is - a better phrasing might be, "all sets are potentially countable." This retains the ability to think of a "current" universe, so that we can say "$X$ is uncountable, but of course $X$ is countable in some forcing extension" - and indeed this matches standard set-theoretic practice, more or less. Sep 11, 2017 at 22:14