Torkel Franzen on truth
in [Logic]
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From: herbzet on 8 Dec 2007 01:45 george wrote: > > On Dec 6, 5:54 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > Oh? Why does it force us to ask that?? Actually I do know Dan, and of > > course we'd both agree with Torkel Franzen's point. > > Oh, stop lying. Oh, come on with the lying already. That really _alienates_ people, which is not what you want. I'm sure you can come up with a different article of obloquy that will be at once sufficiently caustic and cathartic! I have confidence in your resourcefulness. > The whole reason I quoted it is that > Isaacson is directly contradicting TF's point here. -- hz
From: Nam D. Nguyen on 8 Dec 2007 03:41 Herman Jurjus wrote: > Newberry wrote: >> >> There are three possibilities: >> 1) We do not have the foggiest idea if PA is consistent and we will >> never know. Hence we do not know if G�del sentence is true. >> 2) The human mind surpasses any computer >> 3) There exists a formalization of arithmetic that can prove its own >> consistency. >> >> But it is not possible to reject 1 and 3 and at the same time claim >> that the human mind does NOT surpass any machine. > > In the strictest sense, the correct answer is 1). > But: if PA should ever turn out to be inconsistent, then the whole > subject of foundations of mathematics as we now know it will have to be > redone: we assume everywhere in mathematics that the natural number > sequence makes sense, and that it has the PA properties. We even presume > it in definitions of the formal language of FOL. > > In short: 1) is correct: we only -do as if- we know PA is consistent, > and we do so for pragmatic reasons. > I totally agree with you. If PA is inconsistent, it might be so without us knowing why: the inconsistency proof might be quite literally beyond human grasp! (One of the fallacy easy to fall into is we somehow know *all* the finite information!). For what's it worth, if PA is inconsistent, 2+2=4 is still "true": it's just PA wouldn't have any model to contain such truth, simply because it doesn't have any model at all.
From: Newberry on 8 Dec 2007 18:55 On Dec 8, 2:04 pm, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Newberry <newberr...(a)gmail.com> writes: > > The cornerstone of TF's argument is that the consistency of ZFC is > > provable in ZFC + an axiom of infinity, which is no longer manifestly > > true. We hesitate as we go higher up in the chain of theories, hence - > > according to TF - we are not better than any machine because we cannot > > say about an arbitrary system that its Goedel formula is true. Yet he > > is absolutely sure that PA and ZFC are consistent. > > Where did TF claim that he was "absolutely sure that ... ZFC [is] > consistent"? > > I see no evidence that that was his view in the cite you posted > earlier in the thread. He has an entire chapter in his book "Skepticism and Confidence", where he refutes the skeptics. "And given that the axioms of ZFC are so utterly compelling, so obviously true in the world of sets, we can do no better than to adopt these axioms as our starting point. Since the axioms are true, they are also consistent." [p.105] "if the axioms of ZFC are manifestly true, they are obviously consistent" [p.105] But it does not matter if TF is convinced that ZFC is consistent. I claim that there are only three possibilities: 1) We do not know if PA's Goedel sentence is true. So we do not know that the set of truth is productive. 2) The human mind surpasses any machine 3) There axists a formalization of arithmetic that can prove its own consistency Thus far I have not seen any convincing argument that we can reject all three. In fact based on what I have seen I am convinced that we cannot. Are you opting for 1)? You are a second convert. Daryl is now also leaning in that direction. It would make more sense than attempting to argue that we can reject all three, but it is quite a departure from what most logicians believe. Peter Smith will certainly differ.
From: Daryl McCullough on 8 Dec 2007 18:57 Newberry says... >I think you would agree that the consistency of PA is a hard fact not >just an emotional state or a similar psychological phenomenon. The consistency of PA is a hard fact. The fact that I *consider* it to be a hard fact is a psychological phenomenon. It doesn't in any way show that there is something going on that surpasses any machine. We can program a machine to believe that the consistency of PA is a hard fact. It would believe it without the need for proof, just as we do. >It means that no matter how long and in what order we keep generating >theorems we will never derive P & ~P. PA can be literally materialized >as a mechanical system, a machine. We are asking if tis machine can >ever produce P & ~P. No machine can answer that question. That's ridiculous. Of course a machine can answer that question. You can program a machine to answer "yes" when asked "Is PA consistent?". You can program a machine so that if asked why it believes that, it will say "Because the axioms are all manifestly true, and no contradiction can follow from manifestly true axioms." What is it you are saying cannot be done by any machine? -- Daryl McCullough Ithaca, NY
From: LordBeotian on 9 Dec 2007 02:52
<berry(a)pop.networkusa.net> ha scritto >> But it does not matter if TF is convinced that ZFC is consistent. I >> claim that there are only three possibilities: >> 1) We do not know if PA's Goedel sentence is true. So we do not know >> that the set of truth is productive. >> 2) The human mind surpasses any machine >> 3) There axists a formalization of arithmetic that can prove its own >> consistency > > [I assume "PA" in 1) is a 'typo' for "ZFC"] > Well, 3) is obviously true; any inconsistent formalization will do > nicely. > If you meant consistent formalization, you're wrong; one could for > example hold that the human mind is equivalent in power to some finite > extension of ZFC which proves the consistency of ZFC [for > definiteness' sake, let's take the extra axiom to be Projective > Determinacy]. You can't be really definite about it: if the human mind is equivalent to a system than it cannot ever know which system it is equivalent to (by Godel's theorem). |