Both of Gödel's incompleteness theorems can be derived from the undecidability of the Halting problem with a short proof. (This is the last part of a fantastic three-part series. I recommend part 1 and 2 for context. The series is based on Ryan O’Donnell's brilliant course slides.)

A slightly weaker form of Gödel's first incompleteness theorem can be derived from the undecidability of the Halting problem with a short proof. The full incompleteness theorems also have a short proof which is similar to the one for undecidability of the Halting problem. I highly recommend the whole series (or at least the previous part) for context. The proofs are based on Ryan O’Donnell's brilliant course slides.)