Torkel Franzen on truth
in [Logic]
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From: Daryl McCullough on 17 Nov 2007 13:32 Newberry says... > >On Nov 16, 7:57 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: > 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? I already answered that question: >> There >> is no consistent theory T (at least not with ordinary >> first-order logic) that has an axiom saying >> >> (T proves S) -> S -- Daryl McCullough Ithaca, NY
From: Newberry on 17 Nov 2007 17:33 On Nov 17, 10:26 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Nov 16, 8:07 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> I have no idea what you mean by "the manifest truth proof", > >> unless it is just the informal argument: > > >> Every axiom of PA is true. > >> Truth is preserved under logical deduction. > >> Therefore, every theorem of PA is true. > >> A contradiction can't be true. > >> Therefore, PA can't prove any contradictions. > > >> This proof can certainly be formalized. It just > >> can't be formalized in PA. > > >1) We can see that every axiom of PA is true. In you proof it is > >probably just an assumption. > > >2) The proof can be formalized in ZFC but we do not know if ZFC is > >consistent. So it may prove falsehoods. > > So? > > >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. > > >> >Did we put the axiom on a meta-level just to avoid the > >> >contradiction? > > >> We put it in at the meta-level because PA doesn't have > >> a truth predicate. > > >> >BTW, I do not understand this: > >> >S <-> T(#S), where #S means the Godel code of S. > >> >The Goedel number of S is true ... ? > > >> No, T(#S) doesn't say #S is true, it says "#S is the > >> Godel code of a true statement". > > >What does it say then? > > I just said it: T(#S) says "S is the code of a true statement". > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Newberry on 17 Nov 2007 17:42 On Nov 17, 10:32 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 16, 7:57 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > > 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? > > I already answered that question: > > >> There > >> is no consistent theory T (at least not with ordinary > >> first-order logic) that has an axiom saying > > >> (T proves S) -> S > > -- > Daryl McCullough > Ithaca, NY OK, so now we are getting somewhere. By using eiher (Ex)(Px, #(F)) --> F or (Ex)(Px, #(F)) --> T(F) we can perfectly prove either. 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. But there is one problem. This is just as compelling as any other axiom: T(F) --> F
From: Daryl McCullough on 17 Nov 2007 19:47 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
From: Daryl McCullough on 17 Nov 2007 20:01
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. -- Daryl McCullough Ithaca, NY |