Beyond Incompleteness
in [Logic]
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From: Jesse F. Hughes on 24 Jun 2010 13:42 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: >> >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? > > Check any reference. What do you think he proposed? > > "Decidability: there should be an algorithm for deciding the truth or > falsity of any mathematical statement." - Wikipedia, Hilbert's Program Hint: not every sentence in formal logic is a mathematical statement. > idiot > >> 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? >> >> I don't think so... Why, look! David even gave you a clue about why your statement was mistaken! -- "Yeah, I know, it's quantum [computing], and all kind of interesting physics associated with what is to many a mystical word, but I have a B.Sc. in physics, and I know that you're just talking about specialized mechanical devices when you talk about quantum computing." -- James S. Harris
From: BURT on 24 Jun 2010 15:29 On Jun 14, 9:03 am, 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. > > 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 Incompleteness will take millions of years to become complete. Science has a great future. Mitch Raemsch
From: Charlie-Boo on 24 Jun 2010 15:40 On Jun 24, 3:29 pm, BURT <macromi...(a)yahoo.com> wrote: > On Jun 14, 9:03 am, 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. > > > 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 > > Incompleteness will take millions of years to become complete. Science > has a great future. "One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike." - Einstein > Mitch Raemsch- Hide quoted text - > > - Show quoted text -
From: BURT on 24 Jun 2010 15:42 On Jun 24, 12:40 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jun 24, 3:29 pm, BURT <macromi...(a)yahoo.com> wrote: > > > > > On Jun 14, 9:03 am, 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. > > > > 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 > > > Incompleteness will take millions of years to become complete. > Science > > has a great future. > > "One thing I have learned in a long life: that all our science, > measured > against reality, is primitive and childlike." - Einstein > > > > > Mitch Raemsch- Hide quoted text - > > > - Show quoted text -- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - It takes a long time. Mitch Raemsch
From: Jesse F. Hughes on 24 Jun 2010 21:13
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 >> >> 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 |