Beyond Incompleteness
in [Logic]
|
From: Charlie-Boo on 26 Jun 2010 12:34 On Jun 25, 10:13 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > >> > "Decidability: there should be an algorithm for deciding the truth or > >> > falsity of any mathematical statement." - Wikipedia, Hilbert's > >> > Program > > [...] > > > > >> I'll let others decide whether that Wikipedia quote is really an > >> accurate statement. Seems too broad to me, since it would imply that > >> there should be an algorithm deciding the truth or falsity of, say, > >> Euclid's postulate. > > > Why is that "too broad"? Since when was Hilbert humble? > > Whether or not he was humble, he was not stupid. Did you forget the subject of this discussion: Hilbert's Big Mistake? > Now, I don't know > history of mathematics well, so I could be butt-wrong on this, but as I > understand it, the independence of Euclid's postulate was well-known > long before Hilbert's program and hence that there are structures in > which it is true and others in which it is false. > > From my modern perspective, then, it doesn't make much sense to say that > Euclid's postulate is either true nor false. The question is ambiguous because it is incomplete, not giving the exact context of the postulate. Any question expressed as a formal mathematical wff has its context defined - its universal set - and can't be ambiguous. C-B > It seems to me, though I > could be wrong, that the same observation was clear in Hilbert's time. > > Thus, in Hilbert's time, it should have been clear that not ever > mathematical statement really *is* either true or false. > > The most doubtful bits of this are that my own interpretation relies > heavily on model theoretic notions (truth in an interpretation, in > particular) and so post-dates Hilbert. Again, it would be nice if > someone more knowledgeable would say whether the Wikipedia > characterization of decidability really is a feature Hilbert aimed for. > > -- > Jesse F. Hughes > > "As you can see, I am unanimous in my opinion." > -- Anthony A. Aiya-Oba (Poeter/Philosopher)
From: Charlie-Boo on 26 Jun 2010 20:17 On Jun 24, 9:13 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > >> > "Decidability: there should be an algorithm for deciding the truth or > >> > falsity of any mathematical statement." - Wikipedia, Hilbert's > >> > Program > > >> If you can't see the difference between that sentence and what you > >> wrote, well, I reckon I can't help you. > > >> Here's what you wrote: > >> >> >> >1. Every sentence in formal logic can be shown to be true or shown to > >> >> >> >be false. > > >> I'll let others decide whether that Wikipedia quote is really an > >> accurate statement. > > > We all (most of us, anyway) know what the truth is. Hilbert believed > > that you could determine if any given proposition in mathematics is > > true or not. Quibbling over terminology is what people who don't know > > enough to say anything significant do. And when you consider the fact > > that there is no real standard for mathematical terminology, even > > bickering over terminology is meaningless. > > You really think that *any* reasonable interpretation of what you said > is equivalent to Wikipedia's quotation? Could one interpret sentence and statement to mean the same thing? Sounds conceivable. Meaningless conversation. > David corrected your terminology. His correction was reasonable. Your > response was petulant denial, which is par for the course. > > -- > Jesse F. Hughes > > "Most of my research is irreducibly complex." > -- James S. Harris- Hide quoted text - > > - Show quoted text -
From: Jesse F. Hughes on 26 Jun 2010 22:43 Charlie-Boo <shymathguy(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! -- Jesse F. Hughes -- A lesson in meta-honesty -- Baba: Thanks for being honest. Quincy (age 7): I won't be honest next time. And that's more honesty.
From: Charlie-Boo on 26 Jun 2010 22:52 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? > Brilliant save, Charlie! The only way you can substantiate what you said is to say something almost identical! C-B > -- > Jesse F. Hughes > -- A lesson in meta-honesty -- > Baba: Thanks for being honest. > Quincy (age 7): I won't be honest next time. And that's more honesty.- Hide quoted text - > > - Show quoted text -
From: Charlie-Boo on 26 Jun 2010 22:55
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? C-B > -- > Jesse F. Hughes > -- A lesson in meta-honesty -- > Baba: Thanks for being honest. > Quincy (age 7): I won't be honest next time. And that's more honesty.- Hide quoted text - > > - Show quoted text - |