Beyond Incompleteness
in [Logic]
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From: Charlie-Boo on 25 Jun 2010 00:07 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? > > 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 - By what authority?
From: Jesse F. Hughes on 25 Jun 2010 10:13 Charlie-Boo <shymathguy(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. 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. 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: Frederick Williams on 25 Jun 2010 10:45 "Jesse F. Hughes" wrote: > From my modern perspective, then, it doesn't make much sense to say that > Euclid's postulate is either true nor false. 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. It may be a misunderstanding of Hilbert's famous 'In mathematics there is no ignorabimus.' and 'For us there is no ignorabimus' remarks. One doesn't have to have an understanding of model theory al a Tarski or Robinson to know that there are models of geometry in which the fifth postulate is true and models in which it is false: that is 19th century stuff. I hasten to point out that I am not the 'someone more knowledgeable' whom you seek. -- I can't go on, I'll go on.
From: Frederick Williams on 25 Jun 2010 11:31 Frederick Williams wrote: > > "Jesse F. Hughes" wrote: > > > From my modern perspective, then, it doesn't make much sense to say that > > Euclid's postulate is either true nor false. 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. > > It may be a misunderstanding of Hilbert's famous 'In mathematics there > is no ignorabimus.' and 'For us there is no ignorabimus' remarks. One > doesn't have to have an understanding of model theory al a Tarski or > Robinson to know that there are models of geometry in which the fifth > postulate is true and models in which it is false: that is 19th century > stuff. > > I hasten to point out that I am not the 'someone more knowledgeable' > whom you seek. I connection with 'we must know, we will know' it may be worth remarking that Hilbert's tenth problem was to _find_ an algorithm. He did not consider (it seems) that there may not be one, and prior to G\"odel & Turing why should he? -- I can't go on, I'll go on.
From: Charlie-Boo on 26 Jun 2010 07:21
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: > >One way to "extend Rosser 1936" i.e. go beyond incompleteness, is to > >ask what purpose Rosser 1936 serves, and how else can we serve that > >purpose. > > >The answer is, he (like Godel and Smullyan) refuted Hilbert's claims > >that the ideal Mathematical system is possible. > > >How can we refute Hilbert in other ways? > > >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. > I don't think so... > > > > >2. Every sentence in formal logic can be proven or refuted by formal > >logic. > > >3. Formal logic can be shown to be consistent. > > >And how do we formalize this? > > >In CBL: > > >1. TW/YES (The set of true sentences is r.e.) > >2. PR/PR* and DIS/PR* (The sets of theorems and refutations are > >representable.) > >3. -PR,TRUE (Not all sentences are provable.) > > >[ Standard CBL (see postings): TW = true sentences, YES = Programs > >that halt yes, PR = Theorems, DIS = ~Theorems, TRUE = all sentences, > >P(a,b)/Q(a,b) = (eM)(aA)P(A,A)=Q(M,A) ] > > >C-B- Hide quoted text - > > - Show quoted text - |