I've heard a lot of people say that Gödel's proof shows that human intelligence somehow goes beyond what a computer could ever do. It's only ever been articulated to me very badly, though not for want of trying. I agree with the conclusion for other reasons, but I just don't get what the argument pertaining to Gödel's proof is supposed to be.
- The argument can't be that Gödel was capable of actually performing the nitty-gritty of proof where a computer cannot, because it has been proved algorithmically multiple times over the last 30 years.
- The argument can't be that there are statements entailed in the proof that the computer wouldn't be able to also prove, because any statements entailed by Gödel's proof would be entailed by the computers.
- The argument can't be that we can spontaneously use new true-but-unprovable statements as axioms because there's also nothing stopping us adopt any set of axioms we like, whether or not they are true or provable in any given system. The capacity for spontaneity, whilst interesting and no doubt significant for differentiating man from machine, does not seem at all specific to Gödel's proof.
- The argument can't be that an algorithm couldn't think of the idea to attempt the proof, because this also isn't specific to Gödel's proof.
So what is it supposed to be?
Edit: Another way of asking this could be: What is the specific quality of Gödel's very complicated proof that is important for understanding cognition and that can't be demonstrated in another, simpler, way. The answer might be that there is nothing, but so many people insist there is something specific about Gödel's proof that has bearing on cognition that I feel the need to put this question out there.