Torkel Franzen on truth
in [Logic]
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From: Newberry on 14 Nov 2007 00:50 On Nov 13, 9:14 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 13, 1:49 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: > >> To know that theory (PA or M) is consistent we don't necessarily need a > >> "proof". > > >Right. So it means that Lucas and Penrose are right, and Franzen is > >wrong? > > That doesn't follow at all. The question that Penrose asked was: > Is it possible for there to be a computer program P(x) such that > > P(x) halts and returns true > <-> > x is the Godel number of a statement that human > mathematicians can become convinced is an absolutely > unassailable truth > > It is irrelevant whether the set of "unassailable truths" > are provable or not. His point was that the human mind surpasses any machine. If the human mind can comprehend a truth that cannot be formally proven then the human mind surpasses any computer. Example: "if a set of axioms is manifestly true then the theory is consistent" is just as compelling as e.g. "~A, A v B |- B." It is not clear how any machine can prove that a theory based on manifestly true axioms is consistent. At least in this thread we failed to explain how it could be conclusively proven.
From: Bill Taylor on 14 Nov 2007 01:54 > >> Conclusive! Obvious! Who could doubt what one learned as a young boy > >> in Sunday school? > > >I don't know. What does one learn about Peano arithmetic as a young boy in > >Sunday school? > > I'm not sure. But an acquaintance of mine explained how the natural > numbers can be represented using lambda calculus. He told me he > learned it in Church. And don't forget that Kleeneness is next to Godelness! ------------------------------------------------------- Bill Taylor W.Taylor(a)math.canterbury.ac.nz ------------------------------------------------------- And God said Let there be numbers And there *were* numbers. Odd and even created he them, He said to them be fruitful and multiply And he commanded them to keep the laws of induction. -------------------------------------------------------
From: LordBeotian on 14 Nov 2007 07:22 "Newberry" <newberryxy(a)gmail.com> ha scritto >> >So let's confine ourselves to PA for now. We can prove that it is >> >consistent, that is we have proven G. How did we manage to do that >> >without running into a contradiction? We did not simply add Con(T) as >> >another axiom, we proved it. I suppose we proved it in some metatheory >> >M. How do we know that M is consistent? >> >> To know that theory (PA or M) is consistent we don't necessaily need a >> "proof". > > Right. So it means that Lucas and Penrose are rigtht, and Franzen is > wrong? How would you draw this conclusion?
From: LordBeotian on 14 Nov 2007 07:29 "Newberry" <newberryxy(a)gmail.com> ha scritto > His point was that the human mind surpasses any machine. If the human > mind can comprehend a truth that cannot be formally proven then the > human mind surpasses any computer. Every truth can be formally proven. Just take the statement of the truth as an axiom. > Example: "if a set of axioms is manifestly true then the theory is > consistent" is just as compelling as e.g. "~A, A v B |- B." It is not > clear how any machine can prove that a theory based on manifestly true > axioms is consistent. At least in this thread we failed to explain how > it could be conclusively proven. I think ZFC can prove that any theory which has a model is consistent, and also can prove that any theory whose axioms are true in a model is consistent.
From: LauLuna on 14 Nov 2007 09:10
On Nov 13, 1:02 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > Relying on reason, and accepting evident truths, is very reasonable. I completely agree. But, is that a good reason to rely on reason? Regards |