It's great that you wrote a post about this topic! When we talked about this I meant exactly Gödel's Incompleteness Theorems, I just didn't remember the name of this issue. This paradox is indeed strange and as a math layperson I don't really think how to comprehend it. It all comes to the (still philosophical question I think) what mathematics and logic really is? Are they similar in matter with physics laws like the speed of light or the gravity? Are they real thing? Are they platonic ideas? Are another mathematics possible in different/alternate worlds?
I will write about Jung & Pauli relationship soon, I think this topic is fairly associated and these gentleman have also racked their brains on this matter.