What do you make of Godel's proof that Arithmetic is Incomplete?

Question

Thats kurt Godel;And have you tried to follow or criticise his
proof?

Although one might criticise this proof,and one may see its beauty;to
me i cannot have one wiyhout the other.
So i think this proof is subject to criticism,and has been in need of such for the last 70 years. Shame that noone has tried,or rather,been successful(and
i acknowledge the world-of-differance
between the two).

Answer 1

What do you mean when you say that the proof is "subject to criticism"? It is a mathematical proof, and as such is not subject to criticism because all it shows is that a certain set of axioms "force" the truth of a certain conclusion. You can't really criticize Godel's theorem, as it is a theorem. I suppose the only thing you could really do is argue against Church's Thesis (which is unproven) which, if false, would open the possibility that there exist sufficiently powerful formal systems that might not be subject to Godel's incompleteness proof.
The proof is actually not that hard to follow, if you've taken a couple of logic classes or understand formal logic reasonably well; so yes, I have followed it. I have also looked for holes, and there are none.
People far smarter than me have spent more time with Godel's theorem and have not found problems with it, nor have I so it is probably correct. (Unless, of course, someone demonstrates the existence of a mechanical procedure that cannot be simulated by a register machine, but that does not seem likely.)
I also at one point thought that the Godel number of the Godel sentence is problematic because it is nested inside itself so form a "modified" Godel sentence (and this has a different Godel number than the original). But, this isn't a problem because the "original" Godel sentence had a variable as the number representing the proof (and not a specific integer.)

Anyway, have fun with Godel...

Answer 2

I do not criticise Godel in any way. Do you see a valid way we might criticise? IMO it just helps us in seeing the beauty of mathematics.