Torkel Franzen on truth
in [Logic]
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From: Newberry on 8 Nov 2007 01:01 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. Is he saying that a) We are absolutely certain about the truths of PA, even those PA cannot prove b) The consistency of PA can be proven in ZFC c) Therefore we can write a computer program emulating ZFC that can generate the truths of PA d) We are absolutely certain about the truths of ZFC, even those ZFC cannot prove e) There is a theory X in which we can prove the consistency of ZFC f) Therefore we can write a computer program emulating X that can generate the truths of ZFC g) We are not certain about the truths of X ??
From: Peter_Smith on 8 Nov 2007 03:26 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!). Your second quote is misleading: What TF in fact wrote was "Nothing in Gödel's theorem in any way contradicts the view that there is no doubt whatever about the consistency of any of the formal systems we use in mathematics." TF isn't there endorsing the view (as your truncated quotation suggests), he is just pointing out that Godel's theorem doesn't refute it -- a point evidently consistent with the first quote. The third quote you give starts with an emphasized "If" in TF. It is a triviality (any set of truths is consistent!). He is not, at least in those quotations, saying any of (a) to (g).
From: Gc on 8 Nov 2007 04:10 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.
From: Gc on 8 Nov 2007 04:13 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?
From: Peter_Smith on 8 Nov 2007 09:21
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.) |