This is getting fairly recherche now! Nevertheless: I think if one were to apply something of an analogue of Godel's incompleteness theorem to the self-referential universe that you mentioned, the universe would have to refer to an encoded version of **itself**, not merely a copy. It has to be the actual thing (albeit in another form). Is a model, of any fidelity at all, actually the real universe, encoded in another way?

And the incompleteness theorem, as I understand it, would not come up with a contradictory statement, if you were somehow to be able to refer to the actual universe rather than a model: it would say, "You're not going to be able to find out whether this is true or false."

This discussion is somewhat moot, though, is it not, for applying theorems of logic to physics is somewhat fraught: the physical world is not defined well enough for theorems of logic to have validity.

I wonder also about whether those quantities inside the black holes really are fundamentally unknowable, or whether we just don't know about them at the moment. That would make them dissimilar to an unprovable statement, which really is fundamentally unknowable (in the sense that we can say it's true or false from what we know already). The irony here is that the statement might well be true, but we'll never know within the confines of the system as it stands.

So: are those quantities inside black holes a priori unknowable, or just unknown for the time being?