In the presence of Godel's Completeness Theorem, the 2nd Incompleteness Theorem has the following strictly model theoretic interpretation: if there exists any model at all of (say) ZFC, there also exists a model M of ZFC such that M contains no (internal) model of ZFC. That is the direct model-theoretic translation of: if ZFC is consistent, then ZFC + (ZFC is inconsistent) is consistent.
My question(s): Either 1) Does this model theoretic statement possibly enjoy a "purely" model-theoretic proof? or 2) Is there a good reason not to expect a purely model-theoretic proof of this statement?
Since category theory affords a rich context for studying internalization, perhaps I should substitute "category-theoretic" for "model-theoretic."
[As a graduate student years ago I wondered whether "every model of ZFC contains an internal model of ZFC" would simply run directly afoul of the Axiom of Foundation until someone refined my understanding of "internal model" and convinced me that the story couldn't be that simple.]