Torkel Franzen on truth
in [Logic]
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From: Alan Smaill on 17 Jan 2008 11:58 george <greeneg(a)cs.unc.edu> writes: > On Jan 16, 2:24 pm, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> It's consistent with everything on p106 of "G's theorem" >> that TF agrees with you that the existence of any such >> world is "STUPID". > > Well, you have the page and I don't, I wondered why you were so convinced about what TF was claiming. > but I still doubt you here. "any such world" is ambiguous. > I am trying to deny 1 world and he is trying to deny another, > I expect. maybe so, but what's written is still consistent with what I take your opinion to be. >> From the page in question: >> >> ".. we need to distinguish between two things: >> what degree of skepticism or confidence regarding mathematical >> axioms or methods or reasoning is justifiable or reasonable, > > Absolutely NO degree, OBVIOUSLY. he makes the distinction just to say that this aspect is not at issue here. > That he is even entertaining this is, well, tragic & typical; but what > is > even MORE of both of the above is that THAT issue winds up getting > CONFLATED with the SEPARATE issue of TRUTH. > >> and what bearing G�del's theorem has on the matter. Perhaps >> we take a dim view of the claim that we know with absolute >> certainty the truth of, say, the axioms of ZFC, but how >> can we use G�del's theorem to criticise this claim?" > > Via the model existence theorem, obviously. > This is completely easy and straightforward; indeed it is > a matter almost purely of definition. As you are presenting > his argument here it is worse than incoherent -- it is just > the opposite of sound. I'm not claiming that TF is right here; what I'm claiming is that nothing *here* commits him to the view that the axioms of ZFC are true. That was Newberry's claim, and that's the issue as far as I'm concerned. -- Alan Smaill
From: Aatu Koskensilta on 17 Jan 2008 15:26 On 2008-01-17, in sci.logic, george wrote: > But if you choose not to, groundless invective is not relevant. It seldom is. Realising this, why not mend your silly ways? -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: kleptomaniac666_ on 28 Jan 2008 11:41 On Dec 11 2007, 7:50 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > In grasping full second-order PA, we need to get our heads around the > idea not just of sets of numbers but the infinitary idea of arbitrary > infinite sets of numbers. And you might reasonably suppose that that > grasping *that* idea involves grasping more than is required to > understand basic arithmetic. > > Arguably -- famously, Dan Isaacson has argued this, and I've > indirectly defended this in print -- first order PA in fact is a > "natural" theory as it captures just the truths that are accessible to > a reasoner merely in virtue of their understanding of arithmetical > ideas. To grasp other claims formulable in the language of first order > arithmetic as true (e.g. to grasp as true PA Gödel's sentence, or to > grasp as true the arithmetization of Goodstein's theorem) involves > making use of "higher order" concepts, concepts that go beyond what > are needed just to understand basic arithmetic. Actually, to bring up an old point again, although it is perfectly standard and a handy way of talking, from a certain point of view it might be thought strange to call first order PA "arithmetical" and not use the same word for a second order arithmetic statement. In as far as "arithmetical" might be interpreted to mean "about numbers", some true properties of numbers require second order quantifiers to even state them, let alone prove them. Any set of natural numbers has a least element. That is second order and cannot be stated in first order form. You can talk about first order formulas, but then someone could come along and say "no, I really mean any set of numbers whatsoever has a least element". It seems to me hard to deny this property of the natural numbers, and hard to deny that it is as much "about numbers" as any statement possible could be. So in that sense, it is an arithmetical statement. I guess what I am trying to say is, you can go beyond the language of first order arithmetic but still be using perfectly "arithmetical" reasoning, if "arithmetical" is interpreted to mean "about numbers". As for whether, in order for humans to deduce certain first order arithmetic statements, it is not the case that there are first order arithmetic principles which are obvious or intuitive to humans and in principle allow us to do a proof of the theorem, but there are second or higher order principles that do the job, that may well be the case. Note that it is not meaningless to talk about a statement being or not being about numbers. Not every statement of mathematics is such a statement, for example the axiom of choice.
From: kleptomaniac666_ on 28 Jan 2008 11:46
I forgot to add, I don't think there is anything much "special" or "natural" about PA from the point of view of how PA is related to the mathematical objects we call "numbers". It is only special from the point of view of it's relation to the human mind. |