Torkel Franzen on truth
in [Logic]
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From: Newberry on 9 Nov 2007 00:01 On Nov 8, 6:21 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 8 Nov, 09:13, Gc <Gcut...(a)hotmail.com> wrote: > > > > > > > On 8 marras, 11:10, Gc <Gcut...(a)hotmail.com> wrote: > > > > On 8 marras, 10:26, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > On 8 Nov, 06:01, Newberry <newberr...(a)gmail.com> wrote: > > > > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > > > indicates that the human mind surpasses any computer. > > > > > > >> ... 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. << p. 55 > > > > > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 > > > > > >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105 > > > > > > I am not sure that I understand what Franzen is saying. > > > > > The first quote you give, I take it is entirely clear (and correct!). > > > > This I don`t understand: > > > "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." > > > > I have thought: If we assume the consistency of PA we can proof in PA > > > + con(PA) that the gödel sentence of PA being true but not-provable > > > (thus it follows from this that also the con(PA) is not provable from > > > the axioms of PA). And certainly we have an least an informally "idea" > > > that PA is consistent. > > > Oh wait. OK. Now I understand? The point is in GENERAL we don`t have > > an idea if the gödel sentence is true, like in New Foundations? > > Yes, I think that's what TF was after: we'll in general not know > whether T's standard Gödel's sentence is true because we'll not know > whether T is consistent. (Of course, we are usually interested in > theories T which we have pretty good reason to think are consistent, > and we are usually not interested in the other cases! TF is just > reminding us of the general situation.) Do you mean that PA is PROBABLY consistent?
From: Peter_Smith on 9 Nov 2007 03:20 On 9 Nov, 05:01, Newberry <newberr...(a)gmail.com> wrote: > On Nov 8, 6:21 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > On 8 Nov, 09:13, Gc <Gcut...(a)hotmail.com> wrote: > > > > On 8 marras, 11:10, Gc <Gcut...(a)hotmail.com> wrote: > > > > > On 8 marras, 10:26, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > > On 8 Nov, 06:01, Newberry <newberr...(a)gmail.com> wrote: > > > > > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > > > > indicates that the human mind surpasses any computer. > > > > > > > >> ... 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. << p. 55 > > > > > > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 > > > > > > >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105 > > > > > > > I am not sure that I understand what Franzen is saying. > > > > > > The first quote you give, I take it is entirely clear (and correct!). > > > > > This I don`t understand: > > > > "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." > > > > > I have thought: If we assume the consistency of PA we can proof in PA > > > > + con(PA) that the gödel sentence of PA being true but not-provable > > > > (thus it follows from this that also the con(PA) is not provable from > > > > the axioms of PA). And certainly we have an least an informally "idea" > > > > that PA is consistent. > > > > Oh wait. OK. Now I understand? The point is in GENERAL we don`t have > > > an idea if the gödel sentence is true, like in New Foundations? > > > Yes, I think that's what TF was after: we'll in general not know > > whether T's standard Gödel's sentence is true because we'll not know > > whether T is consistent. (Of course, we are usually interested in > > theories T which we have pretty good reason to think are consistent, > > and we are usually not interested in the other cases! TF is just > > reminding us of the general situation.) > > Do you mean that PA is PROBABLY consistent? Read "pretty good reason" to mean "at least pretty good reason, maybe conclusive reason". As it happens I think there are conclusive reasons to believe PA consistent.
From: aatu.koskensilta on 9 Nov 2007 07:17 On 9 Oct, 10:20, Peter_Smith wrote: > Read "pretty good reason" to mean "at least pretty good reason, maybe > conclusive reason". As it happens I think there are conclusive reasons > to believe PA consistent. Yes, PA is obviously consistent. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: abo on 9 Nov 2007 08:00 On Nov 9, 9:20 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > As it happens I think there are conclusive reasons > to believe PA consistent. And: On Nov 9, 1:17 pm, aatu.koskensi...(a)xortec.fi wrote: > > Yes, PA is obviously consistent. Conclusive! Obvious! Who could doubt what one learned as a young boy in Sunday school?
From: Herman Jurjus on 9 Nov 2007 08:20
aatu.koskensilta(a)xortec.fi wrote: > On 9 Oct, 10:20, Peter_Smith wrote: >> Read "pretty good reason" to mean "at least pretty good reason, maybe >> conclusive reason". As it happens I think there are conclusive reasons >> to believe PA consistent. > > Yes, PA is obviously consistent. So what? -- Cheers, Herman Jurjus |