Both of Gödel's incompleteness theorems can be derived from the undecidability of the Halting problem with [a short proof](https://www.youtube.com/watch?v=7fvkbvWaRPk). (This is the last part of a fantastic three-part series. I recommend part [1](https://www.youtube.com/watch?v=YrKLy4VN-7k) and [2](https://www.youtube.com/watch?v=svOTZEbj3ys) for context. The series is based on Ryan O’Donnell's brilliant [course slides](https://www.cs.cmu.edu/~aada/courses/15251s16/www/slides/15251-s16-lecture16.pdf).)