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Diagonalization arguments (which are used in Gödel's theorem) do require some infinity. If there were only a finite number of such "True, but unprovable using

Diagonalization arguments (which are used in Gödel's theorem) do require some infinity. If there were only a finite number of such "True, but unprovable using your current proof rules" sentences, then you could simply add those sentences to your list of proof rules and there would be nothing "True, but unprovable" anymore. For Gödel's theorem to work, you need at least an infinite supply of those kinds of sentences. Also here there's no need to actually write down all of them exhaustively. It's just important to be able to find "yet another one" whenever you would like to, which requires an inexhaustible supply.

cellular 7y ago
"which requires an inexhaustible supply. " But this is how you can stress the theorem. I know it will take a long time, so consider a universe where only 1000 bits of matter actually exist, then try to use the theorem and you won't have enough ink to even hold the theorem, and sets in memory. It's the same for our universe, just with more bits. "you could simply add those sentences to your list of proof rules " Thanks, for responding. Do you mean axioms? I don't understand why that would be a solution. I'll have to read Godel again.
- Hercuros 7y ago
  Yes, strictly speaking that's true, but then if the universe were slightly bigger, or you used a different encoding, you'd get different results. It feels kind of arbitrary to stop thinking about what would happen if the universe contained just one more bit. Philosophically, I think it's also more interesting that Gödel's theorem would continue to produce such sentences, regardless of how large our universe happens to be. You could never have a universe that is so big that it could contain all Gödel sentences, even if our universe were bigger. This is a much more fundamental limitation. Gödel's theorem gives you a sentence S where you can prove neither S nor its negation from the current axioms P. Of course, if you then add S (or its negation) as an axiom, then you CAN prove it. But then Gödel's theorem will just give you another sentence S', ad infinitum. You can never win this game.

cellular 7y ago

"which requires an inexhaustible supply. " But this is how you can stress the theorem. I know it will take a long time, so consider a universe where only 1000 bits of matter actually exist, then try to use the theorem and you won't have enough ink to even hold the theorem, and sets in memory. It's the same for our universe, just with more bits. "you could simply add those sentences to your list of proof rules " Thanks, for responding. Do you mean axioms? I don't understand why that would be a solution. I'll have to read Godel again.

Hercuros 7y ago
Yes, strictly speaking that's true, but then if the universe were slightly bigger, or you used a different encoding, you'd get different results. It feels kind of arbitrary to stop thinking about what would happen if the universe contained just one more bit. Philosophically, I think it's also more interesting that Gödel's theorem would continue to produce such sentences, regardless of how large our universe happens to be. You could never have a universe that is so big that it could contain all Gödel sentences, even if our universe were bigger. This is a much more fundamental limitation. Gödel's theorem gives you a sentence S where you can prove neither S nor its negation from the current axioms P. Of course, if you then add S (or its negation) as an axiom, then you CAN prove it. But then Gödel's theorem will just give you another sentence S', ad infinitum. You can never win this game.