(First, I apologize if I display any fundamental misunderstanding of how set theory works.)
I had a question regarding the limitations of ZFC (assuming its consistency, of course.) Is there any constructable set whose cardinality cannot be determined? As I can't think of a better or more rigorous definition of "constructable," let's just say use "definable in ZFC;" i.e. the real numbers can be constructed, but a hypothetical set which would disprove the continuum hypothesis, while not forbidden by ZFC, cannot be constructed.
For a more rigorous (though slightly less obviously connected) question, do constructable sets A and B exist such that there is an injection from A to B, but it cannot be determined whether there is a bijection?