Torkel Franzen on truth
in [Logic]
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From: Peter_Smith on 11 Nov 2007 14:31 On 11 Nov, 17:46, Newberry <newberr...(a)gmail.com> wrote: > On Nov 11, 12:29 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > On 11 Nov, 03:05, Newberry <newberr...(a)gmail.com> wrote: > > > > On Nov 9, 4:17 am, aatu.koskensi...(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. > > > > OK, how do we reconcile it with this? > > > > >> ... 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 conflict at all between what I said (something TF held > > too), and that latter quote. To hold that PA is clearly consistent is > > quite compatible with holding that, with some arbitrarily thrown- > > together extension of Q, we won't in the general case know whether it > > is consistent, and hence won't know whether its canonical Gödel > > sentence is true. > > 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? Depends what M is: it if is a suitable set theory, by getting your head around the idea of the structure of the iterative hierarchy. (I know this might sound odd coming from someone whose day-job is as a philosopher, but frankly, I do find "how we know?" questions are as entirely boring applied to maths as applied to claims about medium- sized dry goods. Scepticism either way is just uninteresting.)
From: abo on 11 Nov 2007 15:11 On Nov 11, 8:31 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 11 Nov, 17:46, Newberry <newberr...(a)gmail.com> wrote: > > How do we know that M is consistent? > > Depends what M is: it if is a suitable set theory, by getting your > head around the idea of the structure of the iterative hierarchy. > > (I know this might sound odd coming from someone whose day-job is as a > philosopher, but frankly, I do find "how we know?" questions are as > entirely boring applied to maths as applied to claims about medium- > sized dry goods. Scepticism either way is just uninteresting.) Of course it's uninteresting to you. As a rule people like the way that they go and are not interested in reflecting on why they go that way. At the least, there's no profit in it. Aunt Bessie goes to the supermarket every Thursday, and she does not take kindly any suggestions that it might be possible to go on Friday.
From: Peter_Smith on 11 Nov 2007 15:29 On 11 Nov, 20:11, abo <dkfjd...(a)yahoo.com> wrote: > On Nov 11, 8:31 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > On 11 Nov, 17:46, Newberry <newberr...(a)gmail.com> wrote: > > > How do we know that M is consistent? > > > Depends what M is: it if is a suitable set theory, by getting your > > head around the idea of the structure of the iterative hierarchy. > > > (I know this might sound odd coming from someone whose day-job is as a > > philosopher, but frankly, I do find "how we know?" questions are as > > entirely boring applied to maths as applied to claims about medium- > > sized dry goods. Scepticism either way is just uninteresting.) > > Of course it's uninteresting to you. As a rule people like the way > that they go and are not interested in reflecting on why they go that > way. I'm interested in reflecting ... when given good reason to do so.
From: abo on 11 Nov 2007 16:08 On Nov 11, 9:29 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 11 Nov, 20:11, abo <dkfjd...(a)yahoo.com> wrote: > > > > > On Nov 11, 8:31 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > On 11 Nov, 17:46, Newberry <newberr...(a)gmail.com> wrote: > > > > How do we know that M is consistent? > > > > Depends what M is: it if is a suitable set theory, by getting your > > > head around the idea of the structure of the iterative hierarchy. > > > > (I know this might sound odd coming from someone whose day-job is as a > > > philosopher, but frankly, I do find "how we know?" questions are as > > > entirely boring applied to maths as applied to claims about medium- > > > sized dry goods. Scepticism either way is just uninteresting.) > > > Of course it's uninteresting to you. As a rule people like the way > > that they go and are not interested in reflecting on why they go that > > way. > > I'm interested in reflecting ... when given good reason to do so. Ah, well "good reason" is a term which, because of its subjectivity, does not advance matters at all. Aunt Bessie has good reason to go to the supermarket every Thursday; she has always done so.
From: Peter_Smith on 11 Nov 2007 16:23
On 11 Nov, 21:08, abo <dkfjd...(a)yahoo.com> wrote: > On Nov 11, 9:29 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > On 11 Nov, 20:11, abo <dkfjd...(a)yahoo.com> wrote: > > > > On Nov 11, 8:31 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > On 11 Nov, 17:46, Newberry <newberr...(a)gmail.com> wrote: > > > > > How do we know that M is consistent? > > > > > Depends what M is: it if is a suitable set theory, by getting your > > > > head around the idea of the structure of the iterative hierarchy. > > > > > (I know this might sound odd coming from someone whose day-job is as a > > > > philosopher, but frankly, I do find "how we know?" questions are as > > > > entirely boring applied to maths as applied to claims about medium- > > > > sized dry goods. Scepticism either way is just uninteresting.) > > > > Of course it's uninteresting to you. As a rule people like the way > > > that they go and are not interested in reflecting on why they go that > > > way. > > > I'm interested in reflecting ... when given good reason to do so. > > Ah, well "good reason" is a term which, because of its subjectivity, > does not advance matters at all. Aunt Bessie has good reason to go > to the supermarket every Thursday; she has always done so. Why is "good reason" subjective? And as for what Aunt Bessie has to do with the question whether we have good reason to doubt, e.g., the truth of PA, I'm completely stumped! |