Incompleteness in Zfc

The statements of Gödel's incompleteness theorems are famous in mathematics, concerning the foundations of the eld. Whilst most mathematicians have heard of the theorems, few outside of mathematical logic would have gone through the technically sophisticated proofs. Instead of proving the original theorem for the theory of natural numbers and utilizing the notion of consistency, we prove a version of the rst incompleteness theorem for the theory of sets and utilize the notion of correctness, thus allowing for a simpler and technically less demanding proof whilst losing none of the philosophical signi cance of the original. We target an audience of general mathematicians not specializing in mathematical logic, and hope to give a rigorous proof of an important result with relatively less technical demand.