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No. 518
It's not that it's inconsistent, it's that it is either inconsistent or incomplete.
You could prove the inconsistency of ZFC within ZFC by showing a contradiction. Similarly, you could show the incompleteness by proving that a theorem is unable to be proven true or false.
No one has been able to do the first; as far as we know ZFC is consistent. If it weren't, that would be a really, really big deal. Like, rewriting the foundations of mathematics big.
On the other hand, incompletenesses have been shown. There are statements in ZFC that could be true or false without violating any of the axioms.
The only way to get around Goedel's incompleteness theorem is to lose one of the conditions. Either your system is incapable of expressing arithmetic on natural numbers (meaning that counting is impossible) or it is inconsistent (meaning that some statement is both true and false).
Systems unable to express arithmetic on the natural numbers are probably very uninteresting, but you might be able to come up with something novel; that's something I don't know much about. Breaking consistency, on the other hand, means that everything is true and false. Those systems are very uninteresting.
