Torkel Franzen on truth
in [Logic]
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From: LordBeotian on 13 Nov 2007 16:49 "Newberry" <newberryxy(a)gmail.com> ha scritto >So let's confine ourselves to PA for now. We can prove that it is >consistent, that is we have proven G. How did we manage to do that >without running into a contradiction? We did not simply add Con(T) as >another axiom, we proved it. I suppose we proved it in some metatheory >M. How do we know that M is consistent? To know that theory (PA or M) is consistent we don't necessaily need a "proof".
From: kleptomaniac666_ on 13 Nov 2007 19:35 On Nov 13, 3:08 pm, Newberry <newberr...(a)gmail.com> wrote: > On Nov 13, 3:56 am, LauLuna <laureanol...(a)yahoo.es> wrote: > > > > > On 12 nov, 16:50, Newberry <newberr...(a)gmail.com> wrote: > > > > On Nov 12, 3:49 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> > > > wrote: > > > > > On 2007-11-11, in sci.logic, Newberry wrote: > > > > > > On Nov 9, 4:17 am, aatu.koskensi...(a)xortec.fi wrote: > > > > >> Yes, PA is obviously consistent. > > > > > > OK, how do we reconcile it with this? > > > > > Reconcile in what sense? There is no apparent contradiction between Torkel's > > > > explanation concerning... > > > > > ... the mistaken idea that "Gödel's theorem states that in any consistent > > > > system which is strong enough to produce simple arithmetic there are > > > > formulas which cannot be proved in the system, but which we can see to be > > > > true." The theorem states no such thing. As has been emphasized, in general > > > > we simply have no idea whether or not the Gödel sentence of a system is > > > > true, even in those cases when it is in fact true. What we know is that the > > > > Gödel sentence is true if and only if the system is consistent, and that > > > > much is provable in the system itself. > > > > > and the observation that PA is obviously consistent. > > > > There are several issues here. > > > > 1) Isn't exhibiting one theory (PA/ZFC) good enough to establish > > > Lucas's argument? > > > 2) What did TF intend to say by "in general"? Did he mean > > > a) the meta, meta-theories in which we establish the consistency of PA > > > and then ZFC etc. Or did he mean > > > b) alternative theories e.g. Quine's set theory > > > > The problem in a) is that there seems to be an infinite regress. As > > > far as b) chances are that we will be able to establish their > > > consistency just like we established the consistency of PA/ZFC.- Ocultar texto de la cita - > > > > - Mostrar texto de la cita - > > > If I don't misunderstand your query on infinite regress along the > > hierarchy of theories, > > I was mainly asking if I interpreted Franzen correctly. We are sure > that PA is consistent and we prove it in ZFC. We are sure that ZFC is > consistent and we prove it in some metatheory. But we are not sure if > this meatatheory is consistent. Is this what he is saying? > > you are posing an 'ultimately philosophical' > > > question: where does our confidence in PA ultimately stems from? > > > Well, it originates from our confidence in reason, in rational > > evidence. That is what Lotze called 'Selbstvertrauen der Vernunft', > > i.e. reason's confidence in reason. > > > We believe some propositions because we are able to derive them from > > evident truths. We believe evident truths because we rely on reason. > > > We rely on reason for no reason? > > > Regards- Hide quoted text - > > > - Show quoted text - When one says one has proved a theorem "for sure", it means one has proved it from axioms that one is "sure" are true. Consistency does not enter the picture.
From: Newberry on 13 Nov 2007 22:48 On Nov 13, 1:49 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: > "Newberry" <newberr...(a)gmail.com> ha scritto > > >So let's confine ourselves to PA for now. We can prove that it is > >consistent, that is we have proven G. How did we manage to do that > >without running into a contradiction? We did not simply add Con(T) as > >another axiom, we proved it. I suppose we proved it in some metatheory > >M. How do we know that M is consistent? > > To know that theory (PA or M) is consistent we don't necessaily need a > "proof". Right. So it means that Lucas and Penrose are rigtht, and Franzen is wrong?
From: Daryl McCullough on 14 Nov 2007 00:14 Newberry says... > >On Nov 13, 1:49 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: >> To know that theory (PA or M) is consistent we don't necessarily need a >> "proof". > >Right. So it means that Lucas and Penrose are right, and Franzen is >wrong? That doesn't follow at all. The question that Penrose asked was: Is it possible for there to be a computer program P(x) such that P(x) halts and returns true <-> x is the Godel number of a statement that human mathematicians can become convinced is an absolutely unassailable truth It is irrelevant whether the set of "unassailable truths" are provable or not. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 14 Nov 2007 00:27
LordBeotian says... > > >"Daryl McCullough" <stevendaryl3016(a)yahoo.com> ha scritto > >> Well, here's an attempt at describing an informal metatheory that >> captures a lot of human metatheoretic reasoning: >> >> 1. Every axiom of ZFC is true. >> >> 2. For every statement Phi in the language of ZFC, >> Phi <-> Phi is true. >> >> 3. If T is any theory in the language of ZFC, and every >> axiom of T is true, then every theorem of T is true. >> >> This informal theory can prove Con(ZFC) and >> Con(ZFC + Con(ZFC)), etc. And it's all perfectly >> mechanical; you can write a program to work out >> all the consequences of rules 1-3. > >What does it mean "etc." here? Sorry, I thought it was obvious. We can define a sequence of theories T_n as follows: T_0 = ZFC T_{n+1} = that theory whose axioms consist of all the axioms of T(n) plus the additional axiom Con(T(n)) Then the informal theory described can prove Con(T_n) for every n. -- Daryl McCullough Ithaca, NY |