Undecidability [sic] in set theory [per se] 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).
 A: There can be no examples like you are asking for. If a family of statements indexed by natural numbers is 'undecidable' in the sense that there is no recursive procedure for deciding whether the $n$-th statement is true ('true' in whatever model of ZFC we are working in), then infinitely many of those questions must be independent of ZFC.  For, if only finitely many were independent, then the following would be a decision procedure for all the others: Run through all ZFC-proofs until you prove the statement or its negation.
A: If you look at "families" in a more general sense there are many examples. For example, the set of $x \in 2^\omega$ that are the graph of a well ordering of $\omega$. This set is well known to be $\Pi^1_1$ complete, and in particular is not decidable (i.e. lightface $\Delta^0_1$). I think that being well ordered is a purely set-theoretic problem. 
This can also be put into a countable setting by replacing the set above with a slightly less natural set, Kleene's $\mathcal{O}$, of all natural numbers that are indices of computable well-orderings of $\omega$. 
There are many similar examples in descriptive set theory. 
