Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 13 Nov 2007 04:21 On 2007-11-13, in sci.logic, Newberry wrote: > Right. So now the question is how do we reconcile the absolute > certainty that PA is consistent with Goedel's theorem, which says that > the consistency of PA is unprovable. It seems that you just proved it. G�del's theorem does not imply that the consistency of PA is unprovable in any absolute sense, only that there is no formal derivation of "PA is consistent" in PA. > You can prove it in ZFC? First of all I do not know if the ZFC proof > is the same one as the manifest truth proof. In ZFC one would probably just show that the finite von Neumann ordinals are a model of PA. > Secondly, is ZFC consistent? Sure. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 13 Nov 2007 06:20 On 2007-11-13, in sci.logic, Herman Jurjus wrote: > Who says they do? Perhaps they just find it 'too easy' as an answer? Given that the consistency of PA is an obvious triviality it should not be surprising the answer is easy. > Glad that you admit that you don't understand the issue. I do indeed find it utterly baffling people should worry about the consistency of PA, all the while accepting, apparently without any qualms, much more abstract mathematical statements unprovable in PA. There are exceptions, of course, such as Edward Nelson, whose objections and doubts are understandable and interesting, even if totally unrelated to the way the naturals are usually conceived, and the way we usually reason in mathematics. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: LauLuna on 13 Nov 2007 06:56 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, 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
From: Aatu Koskensilta on 13 Nov 2007 07:02 On 2007-11-13, in sci.logic, LauLuna wrote: > We rely on reason for no reason? Relying on reason, and accepting evident truths, is very reasonable. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: abo on 13 Nov 2007 08:52
On Nov 13, 12:20 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: There are > exceptions, of course, such as Edward Nelson, whose objections and doubts > are understandable and interesting, even if totally unrelated to the way the > naturals are usually conceived, and the way we usually reason in > mathematics. What are Nelson's objections and doubts which you understand? As near as I can tell, he complains about the Platonic existence of the naturals and then takes out his doubts on induction, all the while still assuming the existence of said naturals. |