New essay on Goedel's Incompleteness Theorem
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From: John Jones on 24 Sep 2009 14:56 Frederick Williams wrote: > Aatu Koskensilta wrote: >> Frederick Williams <frederick.williams2(a)tesco.net> writes: >> >>> Aatu Koskensilta wrote: >>>> ... >>>> cleared up -- but perhaps, having died over fifty years ago, you no >>>> longer recall Kreisel's explanation... >>> What?! >> John Jones has in the past informed us he's none other than old Witters >> himself. > > Oh, sorry, I didn't know that. Well now you are put straight on the matter. > If I thought I was W's reincarnation I'd > be too ashamed to admit it. > >> Kreisel in some review, article, piece of maundering, recounts >> he once explained G�del's proof to Wittgenstein, in recursion theoretic >> terms, and that Wittgenstein didn't have any problem with the proof >> after that. The infamous passages on the incompleteness theorem were >> written, so one surmises, prior to this. > (etc, etc) Witt would never have gone along with Godel's idea, and it seems he never did. There are a few angles of attack. I draw attention to one angle, mentioned in that paper I quoted, namely that the authors say that Godels proof can be fully and properly implemented yet, by Witts lights, miss its target entirely. And so I have written, thusly, on this noble newsgroup.
From: Newberry on 24 Sep 2009 15:50 On Sep 24, 11:44 am, LauLuna <laureanol...(a)yahoo.es> wrote: > On Sep 22, 11:17 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > > > > > In symbols, > > > G = ~ Pr S [~ Pr S x] > > = ~ Pr [~ Pr S [~ Pr S x]] > > = ~ Pr [~ Pr [~ Pr S [~ Pr S x]]] > > = ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]] > > . . . > > This notation suggests there is a free variable in '~Pr S [~Pr S x]' I also think that he above notation is incorrect. > that can be replaced by the Gödel number of '~Pr S x'. But it is not > so, The argument in that formula is already the Gödel number of that > formula, not a free variable. > > This makes any chain of reference terminate. You have a formula G > that, when metatheoretically interpreted, speaks about the formula G. > Full stop. > > Regards
From: Newberry on 24 Sep 2009 16:28 On Sep 24, 5:39 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Scott H <zinites_p...(a)yahoo.com> writes: > > On Sep 24, 7:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > >> Alternatively, and perhaps more perspicuously, if we include a symbol > >> for every primitive recursive (definition of a) function in the > >> language, the Gödel sentence for T has the form > > >> (x)P(t,x) > > >> where t is a closed term the value of which is, provably in T, the code > >> for G. > > > Well, t is what I meant by G'. I just called it by a different > > variable name. > > But t is not a sentence. It is a closed term in the language of > primitive recursive arithmetic, the value of which is a code for G. In > light of this, how do we make any sense of your suggestion, that > > We must remember, however, that Gödel's theorem is founded not on > self-reference but on endless reference, and that the truth value of G > could turn out to be independent of the truth value of its statement of > reference, G'. Let me add a few comments. I was booted out of here when I suggested that Goedel's sentence was self-referential. Then I was lectured that it is self-referential but not "literally self-referential." It remainds me of Scotts theory that it acts like self-referential when omega-consistency and recursive axiomatizability are added. I do not know how relevant these subtle distinctions are. It seems to me that Goedel's sentence is basically self-referential. [BTW, your frend Torkel Franzen argues that it IS self-referential.] Having said that, self-reference can be unrolled into infinite reference. In fact self-reference is a pre-requisite for infinite reference. So you cannot claim that you have infinite reference but no self-reference. Yet it could be that "the truth value of G could turn out to be independent of the truth value of its statement of reference, G'." but then G would not be equivalent to G'. > > ? > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon mann nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 24 Sep 2009 16:49 Newberry wrote: > On Sep 24, 5:39 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> Scott H <zinites_p...(a)yahoo.com> writes: >>> On Sep 24, 7:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>>> Alternatively, and perhaps more perspicuously, if we include a symbol >>>> for every primitive recursive (definition of a) function in the >>>> language, the G�del sentence for T has the form >>>> (x)P(t,x) >>>> where t is a closed term the value of which is, provably in T, the code >>>> for G. >>> Well, t is what I meant by G'. I just called it by a different >>> variable name. >> But t is not a sentence. It is a closed term in the language of >> primitive recursive arithmetic, the value of which is a code for G. In >> light of this, how do we make any sense of your suggestion, that >> >> We must remember, however, that G�del's theorem is founded not on >> self-reference but on endless reference, and that the truth value of G >> could turn out to be independent of the truth value of its statement of >> reference, G'. > > Let me add a few comments. I was booted out of here when I suggested > that Goedel's sentence was self-referential. Then I was lectured that > it is self-referential but not "literally self-referential." It > remainds me of Scotts theory that it acts like self-referential when > omega-consistency and recursive axiomatizability are added. I do not > know how relevant these subtle distinctions are. It seems to me that > Goedel's sentence is basically self-referential. [BTW, your frend > Torkel Franzen argues that it IS self-referential.] > > Having said that, self-reference can be unrolled into infinite > reference. In fact self-reference is a pre-requisite for infinite > reference. So you cannot claim that you have infinite reference but no > self-reference. For what it's worth, Godel's sentence G is a formula finite in length, whether or not it's "meant" to be self-referential. > > Yet it could be that "the truth value of G could turn out to be > independent of the truth value of its statement of reference, G'." but > then G would not be equivalent to G'.
From: Newberry on 24 Sep 2009 23:10
On Sep 21, 7:33 am, "Scott H" <nospam> wrote: > I give a concise, formal proof and a short intuitive description of ZFC + > ~G. > > http://www.hoge-essays.com/incompleteness.html > > Any constructive feedback is welcome. I think there is something wrong with this [~ Pr S [~ Pr S x]] You say: [QUOTE] This function, which I will denote S, transforms a property into a statement that the symbol for the property has that property. ... For example, the effect of the substitution function on the property "x is prime" is to create the statement, "The statement 'x is prime' is prime." [END OF QUOTE] But 'x' is not a property. So what does "S x" mean? Is 'x' a string, a stand alone free variable, or is it a Goedel number. If it is a Goedel number then it is a specific number and I would not denote it as 'x', which suggests a free variable. |