New essay on Goedel's Incompleteness Theorem
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From: George Greene on 27 Sep 2009 14:34 On Sep 22, 7:56 am, Scott H <zinites_p...(a)yahoo.com> wrote: > Are you sure? Kant believed that even the statement 7 + 5 = 12 is > synthetic. Kant is just old and irrelevant. With respect to the modern paradigm, Kant can't produce anything but cant. And just in case you didn't know, 7+5=12 ISN'T synthetic.
From: Jesse F. Hughes on 27 Sep 2009 15:39 Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes: > David C. Ullrich <dullrich(a)sprynet.com> writes: > >> Aatu may be a real mathematician, but _you've_ been essentially >> ignoring his attempts to "do his job". > > As a mathematician you should know very well logicians aren't real > mathematicians. Perhaps he doesn't think you're a real logician. -- Mo memorized the dictionary But just can't seem to find a job Or anyone who wants to marry "Memorizin' Mo", Someone who memorized the dictionary. Shel Silverstein
From: Marshall on 27 Sep 2009 15:41 On Sep 26, 5:06 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > Yes. I posted this at least a couple of times in the past. But basically, > > for Godel's theorem to work with *all* formal systems (meeting certain > > requirements of course), a reservoir of _infinite_ number of primes would > > be needed, _without_ a constraints as to what value they might be. > > There is certainly an infinite number of primes. I don't know what > "without constraints as to what they might be" means. The set > of primes that I'm familiar with has constraints on it. For example, > it has the constraint that 4 is not a member. This doesn't present > a problem for using primes in encoding statements, however. > > (And anyway, aren't there other possible encoding techniques > besides those using primes? I could be wrong.) Apparently I wasn't wrong. http://en.wikipedia.org/wiki/Proof_sketch_for_Gödel%27s_first_incompleteness_theorem gives an encoding scheme that does not use primes. Still other schemes are possible. Marshall
From: Scott H on 27 Sep 2009 16:53 On Sep 27, 9:47 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Here t is a term, not a sentence "in the model G is about", whatever > that means. I'm afraid your further elucidations are of little help. As > noted, the Gödel sentence of a theory T, in the language of primitive > recursive arithmetic, has the form (x)P(t,x) where t is a term the value > of which is the code for G. I assume that P(t,x) means, "x is not a proof of the arithmoquine of t," and t is a term for the Goedel code of (x)P(y,x), where y is a free variable. If so, then the statement [(x)P(t,x)] is a statement of T, and [(x)P(t,x)] <=> [(x)P([(x)P(t,x)],x)], where [(x)P(t,x)] on the right-hand side is a statement of T'. > What does it mean to say that "the truth > value of G could turn out to be independent of its statement of > reference, G'"? If we added ~G as an axiom to ZFC, then G' would provable while G would not be.
From: Aatu Koskensilta on 27 Sep 2009 16:58
Scott H <zinites_page(a)yahoo.com> writes: > I assume that P(t,x) means, "x is not a proof of the arithmoquine of > t," No, P(t,x) is the formalisation, in the language of primitive recursive arithmetic, of "x is not a proof in T of the sentence with code t", where t is a term the value of which is the code for the sentence (x)P(t,x). > If we added ~G as an axiom to ZFC, then G' would provable while G > would not be. What is G'? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |