Godels theorem ends in paradox-simply proof
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From: Newberry on 29 May 2010 19:55 On May 27, 4:21 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > byron says... > > > > > > > > >it is shown by colin leslie dean that Godels theorem ends in paradox > > >it is said godel PROVED > >"there are mathematical true statements which cant be proven" > >in other words > >truth does not equate with proof. > > >if that theorem is true > >then his theorem is false > > >PROOF > >for if the theorem is true > >then truth does equate with proof- as he has given proof of a true > >statement > >but his theorem says > >truth does not equate with proof. > >thus a paradox > > You are not making any sense. Godel's theorem shows that > not all true statements are provable. Actually Goedel's original theorem proves that there statements S such that neither S nor ~S is provable (if the theory is consistent.) His Theorem VI (or whatever the number was) does not say anythng about truth. > It doesn't say that > *NO* true statements are provable. > > You need to learn this stuff from someone other than the > Australian idiot, Colin Leslie Dean. Goedel was also Australian, right? > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Charlie-Boo on 29 May 2010 23:06 On May 27, 1:15 am, byron <spermato...(a)yahoo.com> wrote: > it is shown by colin leslie dean that Godels theorem ends in paradox > > it is said godel PROVED > "there are mathematical true statements which cant be proven" > in other words > truth does not equate with proof. > > if that theorem is true > then his theorem is false > > PROOF > for if the theorem is true > then truth does equate with proof- as he has given proof of a true > statement In this case, but not in all cases. Godel proves (exists w) ~ ( |-w <=> |=w), not (all w) ~ ( |-w <=> |=w) C-B > but his theorem says > truth does not equate with proof. > thus a paradox
From: herbzet on 2 Jun 2010 22:18 Newberry wrote: > Daryl McCullogh wrote: > > You [byron] are not making any sense. Godel's theorem shows that > > not all true statements are provable. > > Actually Goedel's original theorem proves that there statements S such > that neither S nor ~S is provable (if the theory is consistent.) His > Theorem VI (or whatever the number was) does not say anythng about > truth. That is quite correct. Well done. Of course, it is a short inference from "neither S nor ~S is provable" to "There is a true statement that is not provable" if one assumes, as classical logic does (but intuitionist logic does not) that one of S or ~S is true. Goedel, as we know, was concerned to address people holding either of classical or intuitionist viewpoints. > > It doesn't say that > > *NO* true statements are provable. > > > > You need to learn this stuff from someone other than the > > Australian idiot, Colin Leslie Dean. > > Goedel was also Australian, right? Right -- perhaps they're related?! -- hz
From: Aatu Koskensilta on 4 Jun 2010 12:04 herbzet <herbzet(a)gmail.com> writes: > Of course, it is a short inference from "neither S nor ~S is provable" > to "There is a true statement that is not provable" if one assumes, as > classical logic does (but intuitionist logic does not) that one of S > or ~S is true. G�del's proof establishes that for a consistent formal theory T there's a Pi-1 sentence P that's true but unprovable in T. That is, P has the form "for all naturals n, Q(n)" with Q a decidable predicate, and if T is consistent, Q in fact holds of all naturals and P is unprovable in T. The proof is perfectly constructive -- "logic-free" even, in technical jargon -- so there's no need to invoke classical logic. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: herbzet on 4 Jun 2010 23:37
Aatu Koskensilta wrote: > herbzet writes: > > > Of course, it is a short inference from "neither S nor ~S is provable" > > to "There is a true statement that is not provable" if one assumes, as > > classical logic does (but intuitionist logic does not) that one of S > > or ~S is true. > > G�del's proof establishes that for a consistent formal theory T there's > a Pi-1 sentence P that's true but unprovable in T. That is, P has the > form "for all naturals n, Q(n)" with Q a decidable predicate, and if T > is consistent, Q in fact holds of all naturals and P is unprovable in > T. Right -- if T is w-consistent, then ~P is not provable in T either. > The proof is perfectly constructive -- "logic-free" even, in > technical jargon -- so there's no need to invoke classical logic. I'm unsure of what, if anything, you disagree with in my post. If you're not disagreeing with anything, I'm not sure what you're driving at. Could you elaborate? -- hz |