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This is just an abstract. I think the novelty he perceives is combining all this into a proof of consistency of mathematics. Page 6 of his paper mentions everyt

by stiff 13y ago

This is just an abstract. I think the novelty he perceives is combining all this into a proof of consistency of mathematics. Page 6 of his paper mentions everything you have just written about: https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/edit https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/... This guy spent a few decades working in mathematical logic. He might overestimate the importance of some things, but I would be careful assuming he suddenly stopped understanding simple technical arguments.

zepolud 13y ago
He mentions those in relation to his own pet theories but in no way does he address GP's argument. What his theorem boils down to is: 1. Let A be a formal system capable of describing the natural numbers. 2. Assume A is inconsistent. It follows that there exists such Φ that A ⊢ Φ and A ⊢ ¬Φ 3. ????? 4. A is consistent! At step 3. he appears to conflate A⊢Φ ∧ A⊢¬Φ with Φ∧¬Φ and use it to refute the assumption in 2. But Φ∧¬Φ is not generally true outside the context of A and therefore the refutation does not follow.
- stiff 13y ago
  My understanding is that if proof by contradiction is allowed in A, Φ ∨ ¬Φ is a true sentence in A, so if you try to include the sentence "A is inconsistent" in A, it follows that A ⊢ Φ and A ⊢ ¬Φ, therefore in A Φ ∧ ¬Φ is true, but this contradicts Φ ∨ ¬Φ, therefore A is consistent. In other words, it is impossible to show the inconsistency of any system that includes the law of excluded middle. This is really trivial in the end. Inconsistency is "Φ ∧ ¬Φ", law of excluded middle is "Φ ∨ ¬Φ", so in the assumption of law of excluded middle is hidden the assumption of consistency. That's what he means by The above theorem means that the assumption of consistency is deeply embedded in the structure of classical mathematics Edit: I agree with you in the end, the proof is valid, but the conclusion does not hold in A, but one "level" above, so it does not contradict Goedels theorem. Am I reading this right?

by stiff 13y ago

zepolud 13y ago

He mentions those in relation to his own pet theories but in no way does he address GP's argument. What his theorem boils down to is: 1. Let A be a formal system capable of describing the natural numbers. 2. Assume A is inconsistent. It follows that there exists such Φ that A ⊢ Φ and A ⊢ ¬Φ 3. ????? 4. A is consistent! At step 3. he appears to conflate A⊢Φ ∧ A⊢¬Φ with Φ∧¬Φ and use it to refute the assumption in 2. But Φ∧¬Φ is not generally true outside the context of A and therefore the refutation does not follow.

stiff 13y ago
My understanding is that if proof by contradiction is allowed in A, Φ ∨ ¬Φ is a true sentence in A, so if you try to include the sentence "A is inconsistent" in A, it follows that A ⊢ Φ and A ⊢ ¬Φ, therefore in A Φ ∧ ¬Φ is true, but this contradicts Φ ∨ ¬Φ, therefore A is consistent. In other words, it is impossible to show the inconsistency of any system that includes the law of excluded middle. This is really trivial in the end. Inconsistency is "Φ ∧ ¬Φ", law of excluded middle is "Φ ∨ ¬Φ", so in the assumption of law of excluded middle is hidden the assumption of consistency. That's what he means by The above theorem means that the assumption of consistency is deeply embedded in the structure of classical mathematics Edit: I agree with you in the end, the proof is valid, but the conclusion does not hold in A, but one "level" above, so it does not contradict Goedels theorem. Am I reading this right?