Torkel Franzen on truth
in [Logic]
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From: Newberry on 8 Dec 2007 21:29 On Dec 8, 3:57 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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? We can as well program a machine to say that PA is inconsistent or that it does not know. Is it a pure accident that PA is consistent and we were also programmed to believe that PA is consistent? Are you saying that we were purely arbitrarily programmed to say "PA is consistent, because the axioms are all manifestly true, and no contradiction can follow from manifestly true axioms", and our psychological makeup was arbitrarily shaped such that the argument appears compelling to us? We could as well have been made such that "PA is consistent" would appear absurd to us? Is this what "psychological reason" means?
From: berry on 9 Dec 2007 00:28 On Dec 7, 10:27 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > george wrote: > > On Dec 7, 4:44 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> Well, I'm mainly taking issue with the claim that "we do not > >> have the foggiest idea if PA is consistent". That's not true. > > > Who is claiming that? TF? I never saw him entertain that seriously > > in life. > > The book just says that "in general", right? > > Not for PA specifically, right? > > > In any case there do exist consistency proofs for PA in systems of set > > theory > > that do seem more than just "foggily" sound. So we do have an idea. > > It's just not the kind of idea that deserves any sort of mathematical > > respect. > > > My personal opinion is that mathematical respect flows out of the > > completeness theorem: IF it is inconsistent, THEN THAT MUST be > > provable. > > Why "must", a _subjective_ verb? What happens if such > inconsistency proof is beyond human reach? Why then, as far as we humans are concerned, the system is consistent after all. So there's nothing to worry about. > > > Therefore, the BURDEN of proof rests ALWAYS UPON people > > expressing doubts about consistency. > > But what happens if the theory is genuinely consistent but it's > *impossible* to know that? Under George's philosophy, we assume it consistent by default, since we have found no proof to the contrary.
From: Ross A. Finlayson on 9 Dec 2007 01:33 On Dec 8, 9:28 pm, be...(a)pop.networkusa.net wrote: > On Dec 7, 10:27 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > > > > > george wrote: > > > On Dec 7, 4:44 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > >> Well, I'm mainly taking issue with the claim that "we do not > > >> have the foggiest idea if PA is consistent". That's not true. > > > > Who is claiming that? TF? I never saw him entertain that seriously > > > in life. > > > The book just says that "in general", right? > > > Not for PA specifically, right? > > > > In any case there do exist consistency proofs for PA in systems of set > > > theory > > > that do seem more than just "foggily" sound. So we do have an idea. > > > It's just not the kind of idea that deserves any sort of mathematical > > > respect. > > > > My personal opinion is that mathematical respect flows out of the > > > completeness theorem: IF it is inconsistent, THEN THAT MUST be > > > provable. > > > Why "must", a _subjective_ verb? What happens if such > > inconsistency proof is beyond human reach? > > Why then, as far as we humans are concerned, the system is consistent > after all. So there's nothing to worry about. > > > > > > Therefore, the BURDEN of proof rests ALWAYS UPON people > > > expressing doubts about consistency. > > > But what happens if the theory is genuinely consistent but it's > > *impossible* to know that? > > Under George's philosophy, we assume it consistent by default, since > we have found no proof to the contrary. What if an argument that the assumption that there is truth implies there is a predicate "true", and then in quantifying over objects, they all satisfy it so there is unrestricted comprehension. Then, in building sets, if there is some supertheory of ZFC containing ZFC's axioms as thus truisms, with the unrestricted comprehension over sets in ZFC, then there's a universal collection, a collection of all the elements of ZFC. Yet, then Cantor/Russell etcetera as paradoxes follow. Otherwise there's not a universal quantifier (with considerations of around three different kinds of "universal" quantifiers, indicating various accords or lack thereof with the transfer principle, for any / for each / for every / for all.) Among reasons I think ZF is inconsistent, consider any theory that has as its elements of discourse those elements of ZF (casually referring to ZF as a collection of sets defined by the set-theoretical non- logical/proper axioms), and as well some other elements. Now, quantifying over those elements with "x is a set in ZF", then that collection is the Russell set, so the set containing "x: x is a set in ZF" is the Russell set so ZF contains an irregular set. Otherwise there's no universe (in a broad sense). People seem quick to accept that the Russell set contains unspecified elements but few address the objects of Peano Arithmetic notionally having a similar concern. Consider Burali-Forti, that the order type of ordinals would be an ordinal so there is no collection of all ordinals in ZF, in terms of a difference between "for any", "for each", "for every", and for "all". For each ordinal, its order type is an ordinal, for all ordinals, their order type is not a set. That's basically in distinction of the transfer principle and making shorthand the notion of arbitrarily extended induction, particularly those structures that are only primitively distinguishable among themselves via induction, a memoryless two-step process. It seems those natural objects bootstrap themselves (hoist by their own bootstraps) into a framework where they have a synthetic interface. That is to say, the natural integers form naturally, as a consequence there is infinity, and the universe, and only after where there is the complete framework of all objects is it possible to synthetically distinguish two integers. They do so from nothing. ZF has no universe. In that sense ZF isn't, for example, a Cantorian set theory, where Cantor wanted both a universe and infinite powerset incongruence in his theory. Those two were found incompatible, thus the universe was discarded. There definitely, by definition, specifically, is a universe where the domain of discourse is no other thing. No theory exists in a vacuum. Then, in consideration of which axioms of ZF might be false, I think the axiom of infinity is incorrectly stated, because a variety of fundamental theorems of a set-theoretical infinity, among all possible set theories, would have that infinity is non-well-founded, and in large structures their grandness presupposes their identity. Then, that would lead to the notion that ZF's axiom of regularity is as well so not-necessarily-true: false. There are no universal truths in a theory without the universe, and where there's a universe it's THE universe. Ross -- Finlayson Consulting
From: berry on 9 Dec 2007 02:15 On Dec 8, 5:55 pm, Newberry <newberr...(a)gmail.com> wrote: > > 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]. Then 1) fails by hypothesis, 2) fails since the extension may be simulated by a machine, and 3) fails by Godel's Theorem.
From: tchow on 18 Dec 2007 12:32
In article <vBD9j.29$Tx.18(a)pd7urf3no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >tchow(a)lsa.umich.edu wrote: >> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > >On the basis of model that "sqrt(2) is irrational" is true, of course. You didn't answer my first question directly. Are you convinced that sqrt(2) is irrational? Or are you convinced only of the statement, "in the standard model of the integers, there are no integers m and n such that m^2 = 2 n^2"? Let me assume that the latter formulation is the only one that you assent to. Then let me ask this: Do you believe (note the word "believe" here) the following statement? (*) In the standard model of the integers, there are no integers m and n such that m^2 = 2 n^2. If so, on what basis do you believe (*)? Not on the basis of proof, according to what you say later. So what other basis do you have? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences |