ZFC set theory,first order theory 
Possible Duplicate:
What is the difference between Gödel's Completeness and Incompleteness Theorems?
what is the relationship between ZFC and first-order logic? 

I am a bit confused by a few things that I have read recently. 
I have read that ZFC is a first order theory and that any part of mathematics can be expressed in ZFC. Now I know that first order logic is complete, however this would seem to contradict the incompleteness theorems (with I have a basic understanding of). I was wondering where I have gone wrong?
Thanks very much for any help (sorry for the silly question)

"Complete" means two different things for a logic (such as first-order-logic) versus for a theory in that logic.
A logic is complete iff: Every sentence that has no counterexample-model can be proved.
A theory is complete iff: Every sentence that has no proof-of-its-negation can be proved.
First-order logic is complete in the first sense. ZFC is (assuming it is consistent) incomplete in the second sense -- that is, there are sentences that ZFC neither proves nor disproves. That's completely compatible with the logic being complete; it just means that for each such sentence there are models of ZFC where it is true, and other models where it is false.
