Note: My experience with Gödel's theorem is quite limited: I have read Gödel Escher Bach; skimmed the 1st half of Introduction to Godel's Theorem (by Peter Smith); and some random stuff here and there on the internet. That is, I only have a vague high level understanding of the theory.
In my humble opinion, Gödel's incompleteness theorem (and its many related Theorems, such as the Halting problem, and Löbs Theorem) are among the most important theoretical discoveries.
However, its a bit disappointing to observe that there aren't that many (at least to my knowledge) theoretical applications of the theorems, probably in part due to 1. the obtuse nature of the proof 2. the strong philosophical implications people aren't willing to easily commit towards.
Despite that, there are still some attempts to apply the theorems in a philosophy of mind / AI context. Off the top of my head:
The Lucas-Penrose Argument: which argues that the mind is not implemented on a formal system (as in computer). (Not a very rigour proof however)
Apparently, some of the research at MIRI uses Löbs Thereom, though the only example I know of is Löbian agent cooperation.
These are all really cool, but are there some more examples? Especially ones that are actually seriously considered by the academic community.