Torkel Franzen on truth
in [Logic]
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From: Newberry on 17 Nov 2007 20:28 On Nov 17, 5:01 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 17, 10:26 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> >What I meant is that we can say > >> >(Ex)(Px, #(F)) --> F > >> >leaving the T out. > >> >Yes, it is inconsistent. > > >> No, it's not inconsistent, if you are careful about it. > > >Please explain. You were the one who claimed it was inconsistent in an > >earlier post. > > We talked about a bunch of different things. I don't know > what P is supposed to mean here. Is it the provability predicate? > If so, the provability predicate for what theory? What is F? > Is it just any contradiction? I'll assume it is. > > If you have a theory T1, then you can define a provability > predicate for T1, call it P1(x,y) (meaning "x is a code > for a proof in T1 of a formula whose code is y"). If T1 > is a sound theory (anything it says about arithmetic is > true in the usual interpretation), then the statement > > Ex (P1(x,#F)) -> F > > is a perfectly true sentence. So you can add that as > an axiom to T1 to get a new theory T2. There is no > problem with consistency. T2 can prove G1, the Godel > sentence for T1. But T2 cannot prove G2, the Godel > sentence for T2. You can define a provability predicate > for T2, call it P2, and you can formulate a perfectly > good statement of arithmetic: > > Ex (P2(x,#F)) -> F > > You can add this as an axiom to T2 to get a new theory T3. > And so on. > > This process gives you a way to go from axiomatizable > true theories to more complete axiomatizable true theories. > > Now, if you want to go for the whole ball of wax and > come up with a theory T_ultimate with the following > property: > > T_ultimate proves > Ex (P_ultimate(x,#F)) -> F > > There is no such theory T_ultimate except for an > inconsistent theory. And this is not true.
From: Newberry on 17 Nov 2007 20:31 On Nov 17, 4:47 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 17, 10:32 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> >> There > >> >> is no consistent theory T (at least not with ordinary > >> >> first-order logic) that has an axiom saying > > >> >> (T proves S) -> S > >OK, so now we are getting somewhere. By using eiher > >(Ex)(Px, #(F)) --> F > > I'm sorry, I don't remember what definition you > are using for "P". I assume it's a proof predicate: > P(x,y) means that x is a proof of the statement > whose code is y. But what *theory* > is used to prove y? And what is F? Is it a > false statement, or what? > > >or > >(Ex)(Px, #(F)) --> T(F) > >we can perfectly prove either. > > What is T(F)? > > >But since the former leads to a > >contradiction we opted for the second - Tarski's truth levels. > >But there is one problem: > >T(F) --> F > >is just as compelling as any other axiom. > > The problem here is that you haven't defined what > the heck you are talking about. What is T(F)? > How about defining your terms before using them? > > T(F) --> F > > isn't compelling to me, because I don't even > know what it means. > > -- > Daryl McCullough > Ithaca, NY P(x,y) is a provability predicate F is any wff T(F) is true T(F) --> F is an axiom schema
From: Newberry on 17 Nov 2007 20:54 On Nov 17, 5:01 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 17, 10:26 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> >What I meant is that we can say > >> >(Ex)(Px, #(F)) --> F > >> >leaving the T out. > >> >Yes, it is inconsistent. > > >> No, it's not inconsistent, if you are careful about it. > > >Please explain. You were the one who claimed it was inconsistent in an > >earlier post. > > We talked about a bunch of different things. I don't know > what P is supposed to mean here. Is it the provability predicate? > If so, the provability predicate for what theory? What is F? > Is it just any contradiction? I'll assume it is. > > If you have a theory T1, then you can define a provability > predicate for T1, call it P1(x,y) (meaning "x is a code > for a proof in T1 of a formula whose code is y"). If T1 > is a sound theory (anything it says about arithmetic is > true in the usual interpretation), then the statement > > Ex (P1(x,#F)) -> F > > is a perfectly true sentence. So you can add that as > an axiom to T1 to get a new theory T2. There is no > problem with consistency. T2 can prove G1, the Godel > sentence for T1. But T2 cannot prove G2, the Godel > sentence for T2. You can define a provability predicate > for T2, call it P2, and you can formulate a perfectly > good statement of arithmetic: > > Ex (P2(x,#F)) -> F > > You can add this as an axiom to T2 to get a new theory T3. > And so on. > > This process gives you a way to go from axiomatizable > true theories to more complete axiomatizable true theories. > > Now, if you want to go for the whole ball of wax and > come up with a theory T_ultimate with the following > property: > > T_ultimate proves > Ex (P_ultimate(x,#F)) -> F > > There is no such theory T_ultimate except for an > inconsistent theory. Do you mean that there is no such extension of PA (classical logic with Peano axioms) or do you mean there is no such extension of ANY theory capable of arithmetic. If you mean the later then I think it is not true.
From: LauLuna on 18 Nov 2007 04:22 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. So, I don't think you can take Franzen's position into an indefiniteley delayed skepticism, so to say. Regards
From: LordBeotian on 18 Nov 2007 08:31
"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? 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*. |