New essay on Goedel's Incompleteness Theorem
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From: Newberry on 29 Sep 2009 12:39 On Sep 29, 9:41 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Scott H <zinites_p...(a)yahoo.com> writes: > > Granted, they *might* be the same theory. This, I believe, is a > > question for Platonists. > > Platonism is irrelevant. It's a triviality that any two theories (in > the same language) with the same axioms are the same theory. > > > In the section 'On the Possible Existence of Supernatural Numbers,' > > I describe how T can be extended by adding a proof of G' that is > > inductively inaccessible from outside T, but becomes accessible > > within T by functioning like a variable. > > Your description is just waffle. You write, for example: > > As we have seen, we can consistently add either G or ~G as an > axiom. Many mathematicians have agreed that G is, Platonically > speaking, a true statement, as its falsity would imply its > provability, and therefore its truth. We must remember, however, that > Gödel's theorem is founded 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'. > > It's a mathematical theorem that the Gödel sentence of a consistent > theory is true. There's no Platonism or any philosophising whatever > involved. The Gödel sentence G of a theory refers, in an unproblematic > sense, to a sentence, but that sentence is just G itself. It makes no > sense to suppose the truth value of G might be independent of the > truth value of G. Yes, but ... I still think we can get valuable insights by unrolling the self-reference. Furthermore there are numerous sentences that are classically equivalent to G. Some of them may not be equivalent in some non-classical logic. > You further write: > > Even so, if we added ~G, the consistency of the system in which G' is > formulated would be destroyed. > > Destroyed how? If we add the negation of the Gödel sentence of a > consistent theory T to the theory we simply obtain a consistent theory > that proves a falsity -- there's no apparent destruction. > > If ~G turned out to be true, then neither in arithmetic nor in its > modern extension, Zermelo-Fraenkel set theory, would we be able to > formulate a consistent representation of the theory under study. In a > sense, mathematics would not even be able to reflect upon itself as > an axiomatic system without facing contradictions. > > Here it's obscure what G is. Is it the Gödel sentence of Peano > arithmetic? Of ZFC? In any case, if the Gödel sentence of a theory > turns out to be false this simply means the theory is > inconsistent. There's no need to invoke any reflecting of mathematics > on itself. > > You conclude with this truly baffling passage: > > By adding the supernatural proof x to the theory, however, we would > destroy the consistency of not only the same theory, formulated > within itself, but also of the old theory without the axiom ~G. Near > the beginning of the previous section, I explained that any proof of > G would translate into the very object whose nonexistence it would > prove, effectively proving itself nonexistent in the act of > existing. The same would be true of a proof of G'. In the system of > ~G, x would simply exist, but in the system of G', x would both exist > and not exist. > > This is waffle so fine it stands in no need of any comment. > > > I hope that this idea will make Goedel's theorem and > > omega-inconsistency easier for people to understand. > > This is very unlikely. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 29 Sep 2009 12:39 Newberry <newberryxy(a)gmail.com> writes: > Yes, but ... I still think we can get valuable insights by unrolling > the self-reference. What insights are those? > Furthermore there are numerous sentences that are classically > equivalent to G. Some of them may not be equivalent in some > non-classical logic. Sure, but what of it? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Newberry on 29 Sep 2009 12:44 On Sep 29, 5:22 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Scott H says... > > >On a side note, suppose you defined a truth predicate in T and > >formulated a statement similar to the Liar Statement by reference to > >its reflection in T': > > >L = [~Tr [~Tr S x]] > ><=> [~Tr [~Tr [~Tr S x]]] > ><=> [~Tr L']. > > >This would give us a truth value of L' that differed from that of L, > >would it not? > > If Tr(x) is any formula of first-order logic, then you can > come up with a formula L such that > > L <-> ~Tr(#L) > > where #L means the Godel number of L. There is no L' involved. > > Yes, if Tr(x) were a truth predicate, this would be a contradiction, > which is an argument that there is no truth predicate for arithmetic > definable in the language of arithmetic. This is the case only for theories without truth value gaps. > > -- > Daryl McCullough > Ithaca, NY
From: Scott H on 29 Sep 2009 16:26 On Sep 29, 12:26 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > You will of course look for discussion wherever you like. Peculiar > outbursts like this are best avoided. I did not write in the tone of an outburst, though this world collectively seems to try to give them to me. It comes as a pinch in my heart, followed by auditory hallucinations and, eventually, retching. I'd venture to say that over five years, my suffering has been comparable to being shot one hundred times. And yet people still try to pull my strings, thinking they haven't gone too far.
From: Marshall on 29 Sep 2009 21:28
On Sep 29, 10:13 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > On Sep 29, 9:26 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > Nobody in their right mind cares about Google Groups ratings. > > > Only insane people care what others think of them? That doesn't > > seem right. > > Why would anyone think only insane people care what others think of > them? Did you honestly not understand that I was rephrasing your claim? Whether or not you agree that the rephrasing was faithful to the original, you must have understood that was what I was doing. Didn't you? If you did, then presumably your response was intended to annoy, and no further discussion seems likely to be fruitful. If by some chance you didn't, then read my earlier post as a rephrasing of your quoted statement; a meaningful rebuttal would involve an explanation of why my rephrasing was not faithful to the original. Marshall |