The incompleteness theorems appear mysterious to many people, from sheer confusion of the statements themselves, to wrongfully applying the theorems to scenarios way out proportion, such as (dis)proving the existence of god. It doesn’t help that when actually *learning* about the theorems in a logic course, most details are usually admitted. This is probably not the case at all universities, of course, but I have now personally experienced two different approaches to Gödel’s theorems:

- Spend most of the time on the recursion theory prerequisites to the theorems, without actually covering the theorems themselves, save for the statements;
- Skip the recursion theory and only give an informal argument of the incompleteness theorems without really showing why we should care about recursiveness.

The reason for not giving a full account of the theorems is of course the perennial enemy of lecturers: time. What I’ll try to do in this post is still *not* to give a complete account of the proofs, but try to explain how it all fits together. Fill in the gaps that I’ve at least encountered during my studies, which can then hopefully help others stitch together whichever parts they might have learned throughout their studies. Here we go.