Beyond Incompleteness
in [Logic]
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From: Charlie-Boo on 14 Jun 2010 12:03 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
From: Aatu Koskensilta on 14 Jun 2010 12:11 Charlie-Boo <shymathguy(a)gmail.com> writes: > I believe, where by Formal Logic I mean the system that Hilbert > envisioned: Your beliefs are irrelevant. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: David C. Ullrich on 14 Jun 2010 12:48 On Mon, 14 Jun 2010 09:03:20 -0700 (PDT), Charlie-Boo <shymathguy(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? 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
From: Charlie-Boo on 24 Jun 2010 13:29 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 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... > > > > >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 -
From: Charlie-Boo on 24 Jun 2010 13:36
On Jun 14, 12:11 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > I believe, where by Formal Logic I mean the system that Hilbert > > envisioned: > > Your beliefs are irrelevant. As you can make no relevant distinction between us and presuming you to be honest and believe what you write, all that you write here is irrelevant. C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |