ZFC comes with First-Order Logic and is certainly complete. You are right in that sense. Let me be more specific: Given a sentence that is true in every model

ZFC comes with First-Order Logic and is certainly complete. You are right in that sense. Let me be more specific: Given a sentence that is true in every model of ZFC, you can prove that sentence. This is true because ZFC is a first-order theory, and this is a property of ANY first-order theory. People that are pointing out incompleteness are talking about something else: They are talking about the fact that there are sentences S where neither S nor not S is provable. Additionally, there is a meta reason why S is believed to be true. For example, take S = encoding of "ZFC is consistent". BUT: This sentence S is true in some models of ZFC, and false in others, because otherwise the completeness of ZFC would make it provable / disprovable. So this boils down to the meta-truth being different from truth in a model. Disclaimer: I just recently looked seriously into this stuff, because I was wondering about completeness of a logic I invented. So if the above is wrong, please somebody correct me!