Torkel Franzen on truth
in [Logic]
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From: Nam D. Nguyen on 27 Dec 2007 23:12 tchow(a)lsa.umich.edu wrote: > In article <BdYcj.22608$DP1.12527(a)pd7urf2no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> And since his " we have no reason to believe... >> is true" is a *supporting hypothesis*, one would have no choice but >> interpreting it as the assertion "the negation of Goldbach's conjecture >> is not true". > > Well, if you refuse to take what he said at face value, but insist on > twisting his words into some totally different statement, then there's > no arguing with you. > > I'm satisfied that you admit that you were totally wrong, You're bordering being dishonest. Where did I admit that? > *provided* we > assume that Daryl meant what he said, and that you are correct only if > we forcibly misinterpret Daryl as saying something completely different > from what he actually said.
From: Nam D. Nguyen on 28 Dec 2007 01:44 Nam D. Nguyen wrote: > tchow(a)lsa.umich.edu wrote: >> In article <BdYcj.22608$DP1.12527(a)pd7urf2no>, >> Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >>> And since his " we have no reason to believe... >>> is true" is a *supporting hypothesis*, one would have no choice but >>> interpreting it as the assertion "the negation of Goldbach's conjecture >>> is not true". >> >> Well, if you refuse to take what he said at face value, but insist on >> twisting his words into some totally different statement, then there's >> no arguing with you. >> >> I'm satisfied that you admit that you were totally wrong, > > You're bordering being dishonest. Where did I admit that? > >> *provided* we >> assume that Daryl meant what he said, and that you are correct only if >> we forcibly misinterpret Daryl as saying something completely different >> from what he actually said. What Daryl stated: "So, for example, a proof in PA + the negation of Goldbach's conjecture would not be very convincing, because we have no reason to believe that the negation of Goldbach's conjecture is true." So his hypothesis is: H = "we have no reason to believe that the negation of Goldbach's conjecture is true." His (reasoning) conclusion based on H is: C = "a proof in PA + the negation of Goldbach's conjecture would not be very convincing". Independent of what he actually intended to say, taken on face value his statement's conclusion is, after being stripped from its informality: C' = "(PA + ~GC) is an inconsistent theory due to ~GC" C' is very technical, H is not. So, if C' is "proven" on the basis a non-technical H, then C' is a religion-like conclusion: it has no reasoning basis. If H needs to be translated to a technical assertion (to be a hypothesis), how could it possibly be not equivalent to this following technical statement: H' = "the negation of Goldbach's conjecture is not true" or equivalently: H'' = "~GC is not provable from Q" ? But if so, H' or H'' would be a religion-like statement because there is no proof to it. So overall, the conclusion C or C' has no proof, no basis at all. Why TC didn't see it and reacted the way he did is really surprising to me.
From: herbzet on 28 Dec 2007 01:35 tchow(a)lsa.umich.edu wrote: > (***) In every model of NBG, there do not exist nonzero integers a and b > such that a^2 = 2 b^2. If a/b is a fraction in lowest terms, then a and b are relatively prime, so b won't divide a without remainder (unless b = 1 or b = -1). Exponentiation will not create any new prime factors! -- hz
From: Peter_Smith on 28 Dec 2007 03:46 On Dec 28, 6:44 am, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > What Daryl stated: > > "So, for example, a proof in PA + the negation of Goldbach's conjecture > would not be very convincing, because we have no reason to believe that > the negation of Goldbach's conjecture is true." > > So his hypothesis is: > > H = "we have no reason to believe that the negation of Goldbach's > conjecture is true." > > His (reasoning) conclusion based on H is: > > C = "a proof in PA + the negation of Goldbach's conjecture would not > be very convincing". > > Independent of what he actually intended to say, taken on face value his > statement's conclusion is, after being stripped from its informality: > > C' = "(PA + ~GC) is an inconsistent theory due to ~GC" It is utterly incomprehensible that you read C' into what Daryl said. In any case not-GC is Sigma_1, PA is Sigma_1 complete, so if not-GC, then PA proves it and (assuming PA is consistent) so is PA + ~GC. If Daryl DID hold not-GC (and how you get from his saying that there is no reason to believe non-GC to his assuming it is a mystery), he -- being a sensible chap -- would be *denying* that (PA + ~GC) is inconsistent. But perhaps that's a typo for C" = "(PA + ~GC) is an inconsistent theory to GC But again Daryl, being a sensible chap doesn't say that either. He at most says that, if GC were true (and we have no reason to deny it), the theory PA + ~GC wouldn't be reliably truth-generating. But that OF COURSE doesn't entail that the theory is inconsistent.
From: Daryl McCullough on 28 Dec 2007 09:07
Nam D. Nguyen says... >So his hypothesis is: > > H = "we have no reason to believe that the negation of Goldbach's > conjecture is true." > >His (reasoning) conclusion based on H is: > > C = "a proof in PA + the negation of Goldbach's conjecture would not > be very convincing". > >Independent of what he actually intended to say, taken on face value his >statement's conclusion is, after being stripped from its informality: > >C' = "(PA + ~GC) is an inconsistent theory due to ~GC" No, I never suggested that C' is an inconsistent theory. I said that it is possibly an unsound theory. That is, it can prove false claims. Unsound does not imply inconsistent, as PA + ~Con(PA) shows. -- Daryl McCullough Ithaca, NY |