It would be very surprising if ZF were not consistent. However, the point is that Gödel's Incompleteness Theorem applies to it in exactly the same way as to first-order Peano arithmetic: if the theory in question is consistent, then it cannot prove its own consistency.
If you believe that Peano arithmetic is consistent, then your reason for believing so must transcend Peano arithmetic.
If one believes that ZF set theory is consistent, then one's reason for believing so must transcend ZF.
This doesn't happen with arithmetic because you can enumerate the theorems of arithmetic. If you try to enumerate the theorems of set theory in the same way, you hit a snag once you have sets of cardinality greater than the integers and start having theorems about each element of them.
This isn't true. Your confusing the theorems with the subject that they're intended to be about. Theorems themselves are just finite strings of symbols from a finite (or countable) alphabet. The theorems of ZF can be enumerated in exactly the same way as those of Peano arithmetic. In each case, you start with the axioms (which can be specified very simply) and then systematically write down every possible proof; the last line of each proof is a theorem. This gives you a way to enumerate all theorems of the system. (You can write a computer program to do this.)
The catch is that there's no way to enumerate (via a computer program or algorithm) all the non-theorems.
Your -> You're
Sorry for the typo.
By the way, the reason your argument doesn't work is that you don't "hit a snag once you have sets of cardinality greater than the integers and start having theorems about each element of them." In fact, most of these elements cannot be individually defined via a formula of set theory. If a real number, say, is undefinable in set theory, there is no way to write a sentence of set theory that talks about it specifically. You can write a sentence of set theory that applies to it (because it says, for example, that "every real number has some property"), but you can't write a sentence that distinguishes it from all other real numbers.
The full picture is trickier than this, since you can't define "definability" in set theory, but that's the idea.