Torkel Franzen on truth
in [Logic]
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From: Newberry on 13 Nov 2007 10:02 On Nov 13, 1:21 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > 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. > > Gödel'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. Thanks for the lecture but that does not help us to get around the problem. > > > 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. Is it the same as the manifest truth proof? > > > Secondly, is ZFC consistent? > > Sure. How do we prove ZFC consistency? > > -- > Aatu Koskensilta (aatu.koskensi...(a)xortec.fi) > > "Wovon man nicht sprechen kann, daruber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Newberry on 13 Nov 2007 10:08 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 -
From: Newberry on 13 Nov 2007 10:44 On Nov 11, 5:39 am, LauLuna <laureanol...(a)yahoo.es> wrote: > On Nov 10, 11:38 pm, Newberry <newberr...(a)gmail.com> wrote: > > > > > > > On Nov 10, 12:52 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > On 10 Nov, 16:53, Newberry <newberr...(a)gmail.com> wrote: > > > > > b) The human mind does not surpass a machine > > > > > TF is in favor of b. > > > > > If b is the case then I > > > > wonder how we can construct a machine that can generate all the truth > > > > of PA. > > > > Why should any machine be able to generate all the truths of (the > > > language of) PA? After all, we can't do that either. > > > How can we construct a machine that can generate all the truth of PA > > that we can? > > Good question. > > In 'Inexhaustibility' TF poses the following question: > > It seems that whenever human logico-mathematical reason (HLMR) sees as > evident a set of axioms, it also sees as evident the proposition that > those axioms are consistent (which is a kind of reflection principle). > But, if HLMR is consistent and sufficiently rich, that proposition > does not always follow from those axioms (by Gödel's second theorem). > So, if there is an initial and sufficiently rich set of logico- > mathematical truths that must be included in HLMR and HMLR is closed > under that kind of reflection principle, there is no algorithm > representing human logico-mathematical reason. > > As I interpret TF, he denies the conclusion by alleging > > 1. It might happen that there is no such thing as a definite HLMR. > > 2. Even if HLMR exists, human finiteness precludes the possibility > that it is closed under that reflection principle: humans will > hesitate as things grow increasingly involved. > > TF's position (very akin indeed to Hofstadter's) seems questionable to > me because it fails to recognize the existence of an ideal legality in > human reason, that is different from what humans can actually perform, > and that he, TF, is implicitly invoking while reasoning. > > Nevertheless, I think TF's arguments show clearly why Lucas's and > Penrose's arguments fail. They both are assuming implicitly that: > > A. HLMR is a definite object > B. HLMR is closed under some reflection principle(s). > > Clearly, A and B does not follow from Gödel's theorem. > Where do Lucas and Penrose asume A and B. The issue is that we can conclude with certainty that G is true. (PA is consistent because the axioms are manifestly true.) Thus far no one explained how we could construct a machine that would do the same.
From: MoeBlee on 13 Nov 2007 12:48 On Nov 13, 7:08 am, Newberry <newberr...(a)gmail.com> wrote: > 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? I wouldn't take that as an accurate summary of his view. Rather, he has a main point in his discussion of skepticism. If you go back to read it, it's really difficult to miss what that point is. MoeBlee
From: LordBeotian on 13 Nov 2007 15:49
"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? |