I would like to understand the mechanism of following statement from this answer by Noah Schweber:
We could work in a non-$\omega$ model of ZFC. In such a model, there are sets the model thinks are finite, but which are actually infinite; so there's a distinction between "internally infinite" and "externally infinite."
How a set could be "internally", ie inside appropriate model be regarded as "finite", but be "externally" infinite? (btw, what means "externally"? An "overmodel" containing the submodel which regards this set as finite)
Eg, could somebody present an illuminative "toy example"? I know so far how construct models with "converse" phenomena, ie where eg the naturals $\omega^M$ of the fixed model $M$ (which tautologically trough the lens of this model are seen as countable) appear internally uncountable, see eg here for idea. Thank's to @Alex Kruckman for pointing out the correct phrasing of this phenomenon.
Here I'm also not 100 percently sure what is meant in by "externally" as contrast to internally (= inside fixed model) A guess: A kind of "distinguished overmodel" which contains all ordinals?
But the converse construction that some set is internally finite but externally infinite appears to me to be less plausible/ intuitive. Could somebody elaborate the phenomenon behind?
Best Answer
Remember that "appear finite" just means "has an injection (in the model) into an element of the thing the model thinks is $\omega$." The point is that we can have a "nonstandard $\omega$," with elements which are - externally - infinite (but internally finite by virtue of literally being elements of the model's $\omega$). E.g. any model of ZFC+"ZFC is inconsistent" must be such a model, since it will need to have a "number" which codes a proof of a contradiction in ZFC and no such truly finite number exists (hopefully!).
I think this becomes much simpler if we shift from set theory to arithmetic. Consider a nonstandard model $\mathcal{M}$ of (say) first-order Peano arithmetic, $\mathsf{PA}$. We have the obvious initial segment embedding $i:\mathbb{N}\rightarrow\mathcal{M}$ whose image (by nonstandardness) is proper; any $m\in\mathcal{M}\setminus ran(i)$ is "externally infinite but internally finite." Set theory makes the picture messier by having lots of additional "stuff" going on, but it's ultimately the same.