Torkel Franzen on truth
in [Logic]
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From: abo on 12 Nov 2007 09:18 On Nov 12, 12:56 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > On 2007-11-11, in sci.logic, abo wrote: > > > What is "good reason" to you may not be "good reason" to someone > > else. For instance, a hard-core theist would hold that there is not > > "good reason" to discuss the existence of God. > > An indifferent atheist might also well find discussing the existence of God > somewhat pointless. Regardless of the question of whether "good reason" is > or is not subjective, it remains a rather trivial platitude that people will > in fact be interested in subjecting this or that to scrutiny, reflection, > doubt, only if presented some incentive to, a "good reason" in a more > mundane sense. Somebody, who was in conversation with PS about this subject, asked him a question, "How do you know?" That would seem to be incentive enough to provide at least a modicum of scrutiny or reflection.
From: Newberry on 12 Nov 2007 10:50 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.
From: Aatu Koskensilta on 12 Nov 2007 12:59 On 2007-11-12, in sci.logic, abo wrote: > Somebody, who was in conversation with PS about this subject, asked > him a question, "How do you know?" That would seem to be incentive > enough to provide at least a modicum of scrutiny or reflection. Surely you know the grounds on which we -- Peter, me, Torkel, and so on -- find PA's consistency obvious by now. On the conception that the naturals are obtained from 0 by repeatedly applying the "add-one"-operation the principle of induction Whenever P is a determinate mathematical property of naturals, if 0 has P, and whenever n has P, n+1 also has P, all naturals have P is manifestly true, as is the principle of definition by primitive recursion, that properties definable by primitive recursion are determinate and well-defined in the relevant sense. Combining this observation with the determinateness of properties expressible in the first order language of arithmetic, that is, those obtainable from the primitive recursive properties by means of the usual logical operations, leads immediately to the conclusion that the axioms of PA are all manifestly true, and hence no contradiction follows from them, by the soundness of the rules of inference of first order logic. Now, if someone finds this explanation incomprehensible even after elaborations, illustrations, gentle persuasion, practice, and so on, I'm stumped. There's simply nothing I can do but shrug. Of course, people might be interested in e.g. what can and cannot be proved without appeal to the totality of the successor function, full induction, etc. for perfectly sensible reasons -- often we obtain mathematical information beyond than just that P is true if we know that P is not only true but also provable from these or those (weak) principles -- but connecting such interests to rather elusive and incomprehensible doubts is pointless. -- 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 12 Nov 2007 13:02 On 2007-11-12, in sci.logic, Newberry wrote: > 1) Isn't exhibiting one theory (PA/ZFC) good enough to establish > Lucas's argument? No. > 2) What did TF intend to say by "in general"? He means that in general, if we're presented with an axiomatisable extension of Q we quite literally have no idea whether or not it is consistent, and consequently whether or not its G�del sentence is true or not. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: abo on 12 Nov 2007 13:22
On Nov 12, 6:59 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > On 2007-11-12, in sci.logic, abo wrote: > > > Somebody, who was in conversation with PS about this subject, asked > > him a question, "How do you know?" That would seem to be incentive > > enough to provide at least a modicum of scrutiny or reflection. > > Surely you know the grounds on which we -- Peter, me, Torkel, and so on -- > find PA's consistency obvious by now. On the conception that the naturals > are obtained from 0 by repeatedly applying the "add-one"-operation the > principle of induction > > Whenever P is a determinate mathematical property of naturals, if 0 has P, > and whenever n has P, n+1 also has P, all naturals have P > > is manifestly true, as is the principle of definition by primitive > recursion, that properties definable by primitive recursion are determinate > and well-defined in the relevant sense. Combining this observation with the > determinateness of properties expressible in the first order language of > arithmetic, that is, those obtainable from the primitive recursive > properties by means of the usual logical operations, leads immediately to > the conclusion that the axioms of PA are all manifestly true, and hence no > contradiction follows from them, by the soundness of the rules of inference > of first order logic. > > Now, if someone finds this explanation incomprehensible even after > elaborations, illustrations, gentle persuasion, practice, and so on, I'm > stumped. There's simply nothing I can do but shrug. Of course, people might > be interested in e.g. what can and cannot be proved without appeal to the > totality of the successor function, full induction, etc. for perfectly > sensible reasons -- often we obtain mathematical information beyond than > just that P is true if we know that P is not only true but also provable > from these or those (weak) principles -- but connecting such interests to > rather elusive and incomprehensible doubts is pointless. > "Now, if someone finds this explanation incomprehensible..." Beware when you need to overstate your case to make a point. Obviously I don't find your explanation incomprehensible, but I do find it lacking. It is lacking at the very beginning, in that the entire point is how or why you think you know that you can always "add one". One other thing. Your statement at the end about "connecting such interests to rather elusive and incomprehensible doubts is pointless" is a subjective claim hidden as an oracular assertion about which there can be no dispute. You think it is pointless, no problem with that. You've been to Sunday School, and you've learned what you've been told. Good for you! Still, whether the doubts are indeed pointless or not is an entirely different matter. |