This question probably has a simple answer, but I just don't see it yet so I thought I'd ask it here.
We know that ZFC proves the consistency of PA. Yet, we use PA as the meta-theory to analyze ZFC, under the even stronger hypothesis that ZFC is consistent. Doesn't assuming that ZFC is consistent (in the meta-theory PA) imply then that the meta-theory PA is consistent, contradicting Godel's second incompleteness theorem?
Gödel's second incompleteness theorem only says that a theory (satisfying certain hypotheses) cannot prove its own consistency. There's nothing wrong with working within such a theory and additionally assuming it is consistent; you will just never be able to prove that assumption within the theory itself. So PA can't prove the consistency of PA, but there's nothing wrong with working in PA and additionally assuming the consistency of PA (or something which implies the consistency of PA).
To put it another way, when you assume the consistency of ZFC, that means you are no longer working just in PA: instead, you are working in PA+Con(ZFC). It is perfectly fine if PA+Con(ZFC) can prove Con(PA); you would only have a problem with Gödel if it could prove Con(PA+Con(ZFC)).