By the completeness theorem ZFC is consistent if there is a model where ZFC is true.
What would happen if we found a model for ZFC? Would we be able to stop worrying if ZFC is inconsistent?
Are people trying to find such a model?
Or is there a proof that no such model exists?
My model theory and set theory is weak in general, but I understand basic ideas.
Best Answer
Gödel's second incompleteness theorem tells us that ZFC (if it is consistent) cannot prove that ZFC is consistent. Therefore, ZFC (if it is consistent) cannot prove that there is a model of ZFC, either. Thus we do not have a proof that no model exists, but we do have a proof that trying to find such a model is a pointless pursuit.
If you make stronger assumptions, such as assuming the existence of large cardinals, then you can prove that models of ZFC exist, and you can study them.