Beyond Incompleteness
in [Logic]
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From: Jesse F. Hughes on 27 Jun 2010 09:22 Charlie-Boo <shymathguy(a)gmail.com> writes: > On Jun 26, 10:43 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Charlie-Boo <shymath...(a)gmail.com> writes: >> > On Jun 14, 12:48 pm, David C. Ullrich <ullr...(a)math.okstate.edu> >> > wrote: >> >> On Mon, 14 Jun 2010 09:03:20 -0700 (PDT), Charlie-Boo >> >> >> <shymath...(a)gmail.com> wrote: >> >> >1st. What did Hilbert claim? I believe, where by Formal Logic I mean >> >> >the system that Hilbert envisioned: >> >> >> >1. Every sentence in formal logic can be shown to be true or shown to >> >> >be false. >> >> >> Hilbert claimed this, eh? >> >> >> So to refute Hilbert we only need to point out that the >> >> sentence >> >> >> Ax P(x) >> >> >> cannot be shown to be true and also cannot be shown to be >> >> false? >> >> > P is a variable and so is not amenable to proof rules that would >> > establish its truth or falsity. (That is the principle on which you >> > rely.) That is like pointing out that 3 is not true or provable, and >> > 3 is not false or refutable. Hilbert's comments have nothing to do >> > with ill-formed expressions. >> >> Ah. The string Ax P(x) is an ill-formed expression. >> >> Brilliant save, Charlie! > > You're right of course, I must admit. So how might we refute > Hilbert's Programme, would you say? I'm no expert on Hilbert's Program, but as I recall, the standard refutation is to show that PA cannot prove every true arithmetic statement. This standard refutation may well be based on a popular misunderstanding of Hilbert. I don't pretend to know mathematical history. -- Jesse F. Hughes "So there is some sense in which your work is more akin to a work of mathematics than a banana is." -- Jim Ferry encourages James S. Harris
From: Jesse F. Hughes on 27 Jun 2010 09:19 Charlie-Boo <shymathguy(a)gmail.com> writes: > On Jun 26, 10:43 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Charlie-Boo <shymath...(a)gmail.com> writes: >> > On Jun 14, 12:48 pm, David C. Ullrich <ullr...(a)math.okstate.edu> >> > wrote: >> >> On Mon, 14 Jun 2010 09:03:20 -0700 (PDT), Charlie-Boo >> >> >> <shymath...(a)gmail.com> wrote: >> >> >1st. What did Hilbert claim? I believe, where by Formal Logic I mean >> >> >the system that Hilbert envisioned: >> >> >> >1. Every sentence in formal logic can be shown to be true or shown to >> >> >be false. >> >> >> Hilbert claimed this, eh? >> >> >> So to refute Hilbert we only need to point out that the >> >> sentence >> >> >> Ax P(x) >> >> >> cannot be shown to be true and also cannot be shown to be >> >> false? >> >> > P is a variable and so is not amenable to proof rules that would >> > establish its truth or falsity. (That is the principle on which you >> > rely.) That is like pointing out that 3 is not true or provable, and >> > 3 is not false or refutable. Hilbert's comments have nothing to do >> > with ill-formed expressions. >> >> Ah. The string Ax P(x) is an ill-formed expression. > > And why is it neither provable nor refutable (or is it neither true > nor false), did you say? It is neither true nor false except in an interpretation. It is nonetheless a perfectly well-formed formula. -- "In a world of ideas there should be a place for people who are not experts [...] to talk out their ideas [...] without facing personal insults. And if they are frustrated[...], why should it be a surprise if they end up contacting news organizations, or Noam Chomsky?" --JSH
From: Charlie-Boo on 27 Jun 2010 11:30 On Jun 27, 9:22 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > On Jun 26, 10:43 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Charlie-Boo <shymath...(a)gmail.com> writes: > >> > On Jun 14, 12:48 pm, David C. Ullrich <ullr...(a)math.okstate.edu> > >> > wrote: > >> >> On Mon, 14 Jun 2010 09:03:20 -0700 (PDT), Charlie-Boo > > >> >> <shymath...(a)gmail.com> wrote: > >> >> >1st. What did Hilbert claim? I believe, where by Formal Logic I mean > >> >> >the system that Hilbert envisioned: > > >> >> >1. Every sentence in formal logic can be shown to be true or shown to > >> >> >be false. > > >> >> Hilbert claimed this, eh? > > >> >> So to refute Hilbert we only need to point out that the > >> >> sentence > > >> >> Ax P(x) > > >> >> cannot be shown to be true and also cannot be shown to be > >> >> false? > > >> > P is a variable and so is not amenable to proof rules that would > >> > establish its truth or falsity. (That is the principle on which you > >> > rely.) That is like pointing out that 3 is not true or provable, and > >> > 3 is not false or refutable. Hilbert's comments have nothing to do > >> > with ill-formed expressions. > > >> Ah. The string Ax P(x) is an ill-formed expression. > > >> Brilliant save, Charlie! > > > You're right of course, I must admit. So how might we refute > > Hilbert's Programme, would you say? > > I'm no expert on Hilbert's Program, but as I recall, the standard > refutation is to show that PA cannot prove every true arithmetic > statement. Yes. That is the first conclusion in Godel's 1st Incompleteness Theorem based on Soundness in the introduction to the 1931 paper. The second conclusion was that not all sentences are provable or refutable ("incompleteness".) The theorem based on w-consistency in the body of the paper came next, then the 2nd theorem. Later was Rosser's and Smullyan's versions. Then C-B showed how to shrink all of these proofs to one short English sentence, in at least two different ways, as well as how to generate proofs from simple properties of 3 specific sets of wffs. C-B > This standard refutation may well be based on a popular > misunderstanding of Hilbert. > > I don't pretend to know mathematical history. > > -- > Jesse F. Hughes > "So there is some sense in which your work is more akin to a work of > mathematics than a banana is." > -- Jim Ferry encourages James S. Harris- Hide quoted text - > > - Show quoted text -
From: Charlie-Boo on 27 Jun 2010 11:58 On Jun 27, 9:19 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > On Jun 26, 10:43 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Charlie-Boo <shymath...(a)gmail.com> writes: > >> > On Jun 14, 12:48 pm, David C. Ullrich <ullr...(a)math.okstate.edu> > >> > wrote: > >> >> On Mon, 14 Jun 2010 09:03:20 -0700 (PDT), Charlie-Boo > > >> >> <shymath...(a)gmail.com> wrote: > >> >> >1st. What did Hilbert claim? I believe, where by Formal Logic I mean > >> >> >the system that Hilbert envisioned: > > >> >> >1. Every sentence in formal logic can be shown to be true or shown to > >> >> >be false. > > >> >> Hilbert claimed this, eh? > > >> >> So to refute Hilbert we only need to point out that the > >> >> sentence > > >> >> Ax P(x) > > >> >> cannot be shown to be true and also cannot be shown to be > >> >> false? > > >> > P is a variable and so is not amenable to proof rules that would > >> > establish its truth or falsity. (That is the principle on which you > >> > rely.) That is like pointing out that 3 is not true or provable, and > >> > 3 is not false or refutable. Hilbert's comments have nothing to do > >> > with ill-formed expressions. > > >> Ah. The string Ax P(x) is an ill-formed expression. > > > And why is it neither provable nor refutable (or is it neither true > > nor false), did you say? > > It is neither true nor false except in an interpretation. The lack of an interpretation makes it ill-formed for our purposes. **Required Field Missing** C-B > It is > nonetheless a perfectly well-formed formula. > > -- > "In a world of ideas there should be a place for people who are not > experts [...] to talk out their ideas [...] without facing personal > insults. And if they are frustrated[...], why should it be a surprise > if they end up contacting news organizations, or Noam Chomsky?" --JSH- Hide quoted text - > > - Show quoted text -
From: Frederick Williams on 27 Jun 2010 13:08
Charlie-Boo wrote: > > On Jun 27, 9:19 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > It [Ax P(x)] is neither true nor false except in an interpretation. > > The lack of an interpretation makes it ill-formed Being well-formed or not is a syntactic matter... > for our purposes. .... Oh, you mean for _your_ purposes. I don't doubt that one could give an account of well-formedness that takes into account interpretations, and if one was talking about natural language it might even be necessary, but you are talking about Hilbert's concept of formal language, aren't you? -- I can't go on, I'll go on. |