I'd like to know of any natural undecidable (not independent!) families of questions in the theory of ZF or ZFC where the questions seem naturally to belong to set theory qua set theory.

(This follows up on my question The purview or scope of set theory qua set theory which didn't get much traction.)

I don't know how to formalize the distinction here-- I'd appreciate help there too -- but I wouldn't accept, say, "the word problem in finitely presented groups" because that has nothing directly to do with the sort of things set theorists usually talk about: higher cardinalities, ordinals, filters, elementary embeddings, etc.,

On the other hand, something like an undecidable family of cardinal arithmetic questions would suit me fine (but not if we know the independence of all of them from ZF(C) - that *decides* their status within the theory).