New essay on Goedel's Incompleteness Theorem
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From: Aatu Koskensilta on 24 Sep 2009 07:57 Scott H <zinites_page(a)yahoo.com> writes: > Well, I hoped you'd like it. Your essay is a fine piece of G�del waffling. > If you see an actual error in the essay, feel free to let me know. Apparently you don't consider it an actual error much of your waffling has nothing whatever to do with the mathematical content of the incompleteness theorems. Take again this passage We must remember, however, that G�del's theorem is founded not on self-reference but on endless reference, and that the truth value of G could turn out to be independent of the truth value of its statement of reference, G'. This is just nonsense. The G�del sentence G for a theory T has the form (x)(P(x) --> Q(x)) where P is a (primitive recursive) predicate such that, provably in T, the code for G is the only natural satisfying P. Alternatively, and perhaps more perspicuously, if we include a symbol for every primitive recursive (definition of a) function in the language, the G�del sentence for T has the form (x)P(t,x) where t is a closed term the value of which is, provably in T, the code for G. In a clear sense G refers to a sentence, but that sentence is just G itself. There's certainly no reference to any sentence G' the truth value of which might differ from that of G. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Scott H on 24 Sep 2009 08:34 On Sep 24, 7:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > This is just nonsense. The Gödel sentence G for a theory T has the form > > (x)(P(x) --> Q(x)) > > where P is a (primitive recursive) predicate such that, provably in T, > the code for G is the only natural satisfying P. Alternatively, and > perhaps more perspicuously, if we include a symbol for every primitive > recursive (definition of a) function in the language, the Gödel sentence > for T has the form > > (x)P(t,x) > > where t is a closed term the value of which is, provably in T, the code > for G. Well, t is what I meant by G'. I just called it by a different variable name.
From: Scott H on 24 Sep 2009 08:36 On Sep 24, 4:27 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Scott H wrote: > > My proof is based on the one in Goedel's original manuscript. > > Have you read it? Did you understand what you read? Yes and yes. I'm still looking for a clear definition of recursive axiomatizability and a proof of the translation theorem.
From: Aatu Koskensilta on 24 Sep 2009 08:39 Scott H <zinites_page(a)yahoo.com> writes: > On Sep 24, 7:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> Alternatively, and perhaps more perspicuously, if we include a symbol >> for every primitive recursive (definition of a) function in the >> language, the G�del sentence for T has the form >> >> (x)P(t,x) >> >> where t is a closed term the value of which is, provably in T, the code >> for G. > > Well, t is what I meant by G'. I just called it by a different > variable name. But t is not a sentence. It is a closed term in the language of primitive recursive arithmetic, the value of which is a code for G. In light of this, how do we make any sense of your suggestion, that We must remember, however, that G�del's theorem is founded not on self-reference but on endless reference, and that the truth value of G could turn out to be independent of the truth value of its statement of reference, G'. ? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 24 Sep 2009 10:46
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> We always should learn from what who have gone before us learnt. That >> doesn't necessarily mean they would _always_ be correct, whether or >> not they realized that, or whether or not they realized that but >> wouldn't want to admit it (perhaps for fear of loosing the >> "knowledge"?). > > The relevance of these reflections escapes me. I haven't said anything > about anyone always being right. Your mentioning of a autodidact: > One of the dangers in being an autodidact -- > and I say this as a fellow autodidact -- is that it is often very > difficult to assess with any accuracy whether some idea, some line of > thought, that springs to mind, is likely to have any significance or > interest, from the point of view of the professional researcher; without > feedback from those in the know it's very easy to get stuck on some > apparently brilliant but in reality vacuous insight... doesn't seem to me as striking a balanced one. The pitfall in self- learning to rely on one's own knowledge, when it turns out to be bad, is as equally grave as when relying on, respectfully speaking, a professional opinion in the field. In mathematics and reasoning, no one is above the possibility of being (inadvertently) wrong, especially when it comes to the issues of foundation. |