The basic result of the incompleteness proofs is pretty simple. The proofs are complicated because the idea was literally unthinkable until programming was invented. The heroic efforts of Gödel, Skolem, Post, and Church to create a mathematics in which doing mathematics
Gödel’s Lost Letter
Gödel’s perspective in this letter is a refutation of the simplistic take on Turing’s proof seen so often in the computer science literature. In particular: After all, one would simply have to choose the natural number n so large that