Beyond Incompleteness
in [Logic]
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From: Charlie-Boo on 27 Jun 2010 13:26 On Jun 27, 1:08 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > 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... The interpretation maps the syntax into something that is true or false. > > for our purposes. > > ... Oh, you mean for _your_ purposes. You're the one who said your statement is true because you have no interpretation (as I predicted, by the way.) > 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? That just means it isn't necessarily necessary. C-B > -- > I can't go on, I'll go on.
From: Frederick Williams on 27 Jun 2010 13:42 Charlie-Boo wrote: > You're the one who said your statement ... I think you're confusing me with someone else. -- I can't go on, I'll go on.
From: Jesse F. Hughes on 27 Jun 2010 15:23 Charlie-Boo <shymathguy(a)gmail.com> writes: > 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. Yes, CBL is a great advance in logic. I've always said so. Congrats! -- "[Y]ou never understood the real role of mathematicians. The position is one of great responsibility and power. [...] You people have no concept of what it means to be an actual mathematician versus pretending to be one, dreaming you understand." -- James S. Harris
From: Jesse F. Hughes on 27 Jun 2010 15:20 Charlie-Boo <shymathguy(a)gmail.com> writes: > 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** This is pointless. -- "So now you see a math person coming out to talk about *his* program which is fast as he says it can count over 89 billions primes in less than a second. How is that objective? It's childish." -- James S. Harris, on objective facts.
From: Charlie-Boo on 28 Jun 2010 09:13
On Jun 27, 3:23 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > 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. > > Yes, CBL is a great advance in logic. I've always said so. > > Congrats! Thanks. And BTW do you know what would be cool? If sarcasm were replaced by specific valid mathematical points. C-B > -- > "[Y]ou never understood the real role of mathematicians. The > position is one of great responsibility and power. [...] You people > have no concept of what it means to be an actual mathematician versus > pretending to be one, dreaming you understand." -- James S. Harris |