[dandanua]

It's ok to use logic to prove theorems in logic. We also use thinking to deduce some facts about our thinking. Self-reference is not always paradoxical. In a higher-order logic some self-referential definitions are legitimate.

[erichahn]

Yes and then you can do self-referential statements again which leads you to the conclusion that Gödels theorems are provable only in a system that cannot prove its own consistency.

[dandanua]

True, but it's not a logical paradox. The fact that a system can't prove its own consistency doesn't imply that this system is inconsistent. The fact that Godel's theorems are provable only in such systems also doesn't imply that they are wrong.

From the common sense the whole situation looks paradoxical, indeed. To prove consistency of some theory we have to use some stronger theory, to prove consistency of that theory we need even stronger theory and so on.

Perhaps, the best way to realize why there is no paradox is the following:

Our memory is finite. As well as our thinking. But the total number of true facts about mathematics is infinite. By constructing theories we are trying to compress the infinite number of true facts into some finite form. Godel's theorem says it's impossible. And it looks quite natural from this perspective.

[erichahn]

I never said it is a paradox. I never said that PA or Q is inconsistent. I said that they cannot prove their own consistency by Gödels theorem. Hence we don't know if Gödels theorem was formalized in an inconsistent system.

Honestly it would be weird if the incompleteness theorems don't apply to themselves.

[ProfHewitt]

The logical contradiction is that allowing the [Gödel 1931]

proposition I'mUnprovable into foundations makes the

foundations inconsistent.