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> An important aspect of the incompleteness theorems, relevant to the present topic, is that one cannot deterministically locate those particular true claims th

by evanb 13y ago

> An important aspect of the incompleteness theorems, relevant to the present topic, is that one cannot deterministically locate those particular true claims that cannot be proven true, to which the theorems apply. If this were not so, it would undermine the very notion of indeterminacy. Right, I agree, but that doesn't necessarily mean that we cannot prove any true claims. Unless I misunderstand, it just means that true claims whose proof we do not know cannot be distinguished from true claims whose proof does not exist. We can search for proofs to both, and sometimes we will find a proof, and sometimes we won't. But not finding a proof doesn't mean that the claim is a Gödel sentence, it could also mean that we just didn't look hard enough. Also: I should really... proof read my posts better. "too hard about apply..." I cringe at the fact that my carelessness will be preserved for posterity.

lutusp 13y ago
> Right, I agree, but that doesn't necessarily mean that we cannot prove any true claims. That's certainly true, but the fact that we can't locate a demarcation between provable and unprovable makes all analysis inside system X problematical as to the true nature of X, with an obvious connection to the present topic. > Unless I misunderstand, it just means that true claims whose proof we do not know cannot be distinguished from true claims whose proof does not exist. Not exactly. The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. The second category is a problem that undermines any global analysis of logical systems. Remember that Russell and Whitehead's famous tome (http://en.wikipedia.org/wiki/Principia_Mathematica http://en.wikipedia.org/wiki/Principia_Mathematica) wasn't edited after Gödel's work, to simply remove those parts that were unsupportable, it was instead abandoned in its entirety. Gödel's work demonstrated that any such deterministic analysis of logical systems wasn't possible, in whole or in part. But again, this doesn't argue that a given logical system can't be used to prove theorems. It only argues against the claim that such a system can be proven to be both complete and internally consistent.
evanb 13y ago
>The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. Can you clarify that a little bit? It seems like you define the demarcation in terms of itself.
- lutusp 13y ago
  Sorry -- I meant that one side of the demarcation line are theorems that we can prove (and prove that we can prove). The other side are two entities -- theorems that are true but not provable, and theorems we don't know enough about to classify. I emphasize that the idea of a demarcation line is itself a matter of debate. Some theorems, by virtue of their unprovability, affect others and cast them into question, which blurs the very notion I'm discussing. I shouldn't have made the demarcation line seem so certain. Even saying that such a thing exists risks giving it more credence that it may deserve -- except as a matter of discussion.

by evanb 13y ago

lutusp 13y ago

> Right, I agree, but that doesn't necessarily mean that we cannot prove any true claims. That's certainly true, but the fact that we can't locate a demarcation between provable and unprovable makes all analysis inside system X problematical as to the true nature of X, with an obvious connection to the present topic. > Unless I misunderstand, it just means that true claims whose proof we do not know cannot be distinguished from true claims whose proof does not exist. Not exactly. The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. The second category is a problem that undermines any global analysis of logical systems. Remember that Russell and Whitehead's famous tome (http://en.wikipedia.org/wiki/Principia_Mathematica http://en.wikipedia.org/wiki/Principia_Mathematica) wasn't edited after Gödel's work, to simply remove those parts that were unsupportable, it was instead abandoned in its entirety. Gödel's work demonstrated that any such deterministic analysis of logical systems wasn't possible, in whole or in part. But again, this doesn't argue that a given logical system can't be used to prove theorems. It only argues against the claim that such a system can be proven to be both complete and internally consistent.

evanb 13y ago

>The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. Can you clarify that a little bit? It seems like you define the demarcation in terms of itself.

lutusp 13y ago
Sorry -- I meant that one side of the demarcation line are theorems that we can prove (and prove that we can prove). The other side are two entities -- theorems that are true but not provable, and theorems we don't know enough about to classify. I emphasize that the idea of a demarcation line is itself a matter of debate. Some theorems, by virtue of their unprovability, affect others and cast them into question, which blurs the very notion I'm discussing. I shouldn't have made the demarcation line seem so certain. Even saying that such a thing exists risks giving it more credence that it may deserve -- except as a matter of discussion.