Torkel Franzen on truth
in [Logic]
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From: LordBeotian on 18 Nov 2007 08:35 "LordBeotian" <pokipsy76(a)yahoo.it> ha scritto >> There is no problem >>> having a truth predicate for arithmetic in a language that >>> goes beyond arithmetic. But you can't have a truth predicate >>> for arithmetic *within* arithmetic. >> >> If you say "can't" what does the impossibility consist of. Is it >> somehow inherently impossible or it "can't" be done because a >> contradiction would result? > > The reason is that if P is a theory including an axiom saying "(P proves > S) -> S" then the statement "(P proves S) -> S" is *provably false*. Ops, I think I have answered to another question... :|
From: LordBeotian on 18 Nov 2007 10:56 "Newberry" <newberryxy(a)gmail.com> ha scritto > There is no problem >> having a truth predicate for arithmetic in a language that >> goes beyond arithmetic. But you can't have a truth predicate >> for arithmetic *within* arithmetic. > > If you say "can't" what does the impossibility consist of. Is it > somehow inherently impossible or it "can't" be done because a > contradiction would result? 1) Truth, despite provability, is not a sintactic property. There is no reason to expect that there exist an algorithm that will say if a number is the godel number of a true arithmetical wff. 2) Actually the assumption that such an algoritm exist lead to a contraddiction.
From: Newberry on 18 Nov 2007 11:44 On Nov 18, 1:22 am, LauLuna <laureanol...(a)yahoo.es> wrote: > On Nov 13, 4:08 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? > > Not, exactly. I don't think he contends the ONLY way we have to > convince ourselves of PA's consistency is by means of ZFC or any other > theory. > > I think he means PA reveals itself evidently consistent after just a > careful examination of its axioms, that it is a mere question of > intuitive evidence. That is exactly what I meant. So there is no conclusive formal proof of PA's consistency. There is an intuitive proof, which is just as compelling as a formal proof. This intuitive proof is not formalizable. It seems to follow that the human mind surpasses any machine. So I am not exactly sure why he is claiming otherwise. He makes a big deal out of G <--> Con(T) and keeps repeating that we cannot conclude about a given theory that its senetnce G is true; we can merely say that G is true if T is consistent. But then he ends up by saying that he is convinced of PA's consistency more than anyone. You can say that you can construct a machine that will prove T(G) - just program the machine with ZFC. But I do not know if it is equivalent to the human mind. For one we do not know if the machine will start producing falsehoods since we do not know if ZFC is consistent. Secondly we are sure of PA's consistency based on the intutive proof. But we are not sure of it consistency based on the ZFC proof. So it seems we are not the machine programmed with ZFC. BTW, how would you formalize the self-evident "truth cannot be inconsistent with itself"? > So, I don't think you can take Franzen's position into an > indefiniteley delayed skepticism, so to say. > > Regards- Hide quoted text - > > - Show quoted text -
From: Aatu Koskensilta on 19 Nov 2007 14:00 On 2007-11-14, in sci.logic, george wrote: > On Nov 12, 6:37 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> > wrote: >> Given that the quoted passage is perfectly clear > > No, it isn't. What do you find unclear in any of the passages quoted? From your comments below any unclarity seems to stem from your idiosyncratic doubts about the notion of truth when applied to arithmetical statements (even of restricted logical complexity). General worries and conundrums about truth, such as the liar, are irrelevant to such applications. -- 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 19 Nov 2007 14:03
On 2007-11-14, in sci.logic, george wrote: > Are you saying that PA continues to remain obviously consistent > even in the context where it is insisted that N is a proper classm and > just plain can't be a set? Sure. No sets or classes need be involved in the observation that PA is obviously consistent. Of course, if we do not assume that infinite sets exist, it can no longer be established that a theory is consistent just in case it has a model on the standard definition of "model". -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |