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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?

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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.

A model is countable if, well, it's countable: if its underlying set is a countable set. Note that internally, of course, a model of ZFC won't think that it is countable, but I'm talking about actual countability, and there are (under reasonable assumptions) countable models of ZFC.

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  • $\begingroup$ But where does this function from a given "countable" model to the naturals live? Clearly it's not in our model. And doesn't this mean that whether a given model is countable depends on whether we accept certain large cardinal properties? Why couldn't we make every model "countable" by viewing it as being within a sufficiently large model? $\endgroup$
    – exfret
    Sep 11, 2017 at 22:00
  • $\begingroup$ @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). $\endgroup$
    – Noah Schweber
    Sep 11, 2017 at 22:05
  • $\begingroup$ As potential clarification the reason why I am saying this is because we can make some sets which any countable model thinks is uncountable as countable by including functions from the countable model to the natural numbers. So why can we say there aren't such functions for an "uncountable" model since it's just a set itself? $\endgroup$
    – exfret
    Sep 11, 2017 at 22:06
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    $\begingroup$ Another approach - not formalism, but still rejecting the idea of a "true" universe of sets - is something like a multiverse view. There are many types of multiverse. In some, there are indeed no "absolutely uncountable" things; see page 25 of the above-linked document. The context for this isn't really large cardinals, but forcing - in particular, the Levy collapse, and its class-sized version. $\endgroup$
    – Noah Schweber
    Sep 11, 2017 at 22:07
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    $\begingroup$ @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. $\endgroup$
    – Noah Schweber
    Sep 11, 2017 at 22:14

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