Torkel Franzen on truth
in [Logic]
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From: Peter_Smith on 21 Nov 2007 10:27 It would take too long to separate out the various confusions in this thread from the occasional bits of informed good sense. But it might be worth pointing out to anyone who is actually *seriously* interested in these issues -- rather than in just sounding off on the basis of half-understood ideas -- that there is actually an extensive literature on exactly what happens when you add truth-theories of different strengths to arithmetic. Start by googling for work by e.g. Volker Halbach or Jeffrey Ketland (for fairly accessible treatments). Worth doing, as some of results -- e.g. about the proof-theoretic strength of what can look to be modest augmentations of first-order PA -- are moderately surprising.
From: Newberry on 22 Nov 2007 01:10 On Nov 21, 6:23 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 20, 8:58 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> Well, there is no such formula that means "F is true" > >> for *arbitrary* F. You can restrict yourself to a specific > >> language L and introduce a predicate T(F) with the interpration > >> that T(F) holds if F is a true formula of language L, but > >> as proved by Tarski, the predicate T cannot be in the language > >> L. > > >I never said it was in the language of L. It is in the meta-langage. > > Well, in the meta-language the schema is just plain vaccuously > true. > > >For any sentence in any theory "if a sentence F is true then F" is > >intuitively self-evident. > > I would say that it is true by definition. "F is true" means the > same thing as F. But now I've lost track of what you are claiming > follows from this schema. I am claiming that by proving that G is true we have proven G. If we have proven this in a metalanguage then this system has its own G_1 and its consistency is unprovable other than in even higher mata- language. Therefore we have no consistency proof of PA at all. Then there is the manifest truth proof: The axioms of PA are manifestly true PA derivations preserve truth Truth cannot be inconsistent with itself i.e. PA is consistent This proof is absolute and conclusive. It means that it is not formalizable. For if it were it would either not be conclusive just like in the meta-theory above or we would have an inconsistency just as if we used P_universal(x,y). It means that the human mind surpasses any machine. How did we manage to surpass any machine? Since the proof is conclusive we must have accomplished some non-formalizable analogy of P_universal. Since it is not formalizable we are off the hook. We can eat the turkey and have it. We have proven G without running into a contradiction. You know what I am getting at?
From: Aatu Koskensilta on 22 Nov 2007 10:00 On 2007-11-20, in sci.logic, Newberry wrote: > Even Aatu Koskensilta posted a contribution not to long ago where he > concedes that the 2-nd theorem depends on the particulars of the > formal system. Of course it does: the second incompleteness theorem applies to theories satisfying the Hilbert-Bernays-Loeb derivability conditions. This has nothing to do with your confused ramblings on Lucas's and Penrose's arguments. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Newberry on 22 Nov 2007 12:20 On Nov 21, 7:27 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > It would take too long to separate out the various confusions in this > thread from the occasional bits of informed good sense. But it might > be worth pointing out to anyone who is actually *seriously* interested > in these issues -- rather than in just sounding off on the basis of > half-understood ideas -- that there is actually an extensive > literature on exactly what happens when you add truth-theories of > different strengths to arithmetic. Start by googling for work by e.g. > Volker Halbach or Jeffrey Ketland (for fairly accessible treatments). > Worth doing, as some of results -- e.g. about the proof-theoretic > strength of what can look to be modest augmentations of first-order PA > -- are moderately surprising. Do they deal with this? QUOTE: Ex (P1(x,#F)) -> F Ex (P2(x,#F)) -> F Ex (P_ultimate(x,#F)) -> F There is no such theory T_ultimate except for an inconsistent theory. END OF QUOTE One thing is apparent about these theories: They are either inconclusive or inconsistent.
From: Peter_Smith on 22 Nov 2007 14:23
On 22 Nov, 17:20, Newberry <newberr...(a)gmail.com> wrote: > On Nov 21, 7:27 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > It would take too long to separate out the various confusions in this > > thread from the occasional bits of informed good sense. But it might > > be worth pointing out to anyone who is actually *seriously* interested > > in these issues -- rather than in just sounding off on the basis of > > half-understood ideas -- that there is actually an extensive > > literature on exactly what happens when you add truth-theories of > > different strengths to arithmetic. Start by googling for work by e.g. > > Volker Halbach or Jeffrey Ketland (for fairly accessible treatments). > > Worth doing, as some of results -- e.g. about the proof-theoretic > > strength of what can look to be modest augmentations of first-order PA > > -- are moderately surprising. > > Do they deal with this? It isn't my job to make up for your not keeping up with the literature and trying to make it up as you go along. Read the stuff and do the work. |