New essay on Goedel's Incompleteness Theorem
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From: Aatu Koskensilta on 29 Sep 2009 12:26 Scott H <zinites_page(a)yahoo.com> writes: > This is a message not just to George Greene, but to all of sci.math > and sci.logic, particularly the member who keeps one-starring my > posts. Nobody in their right mind cares about Google Groups ratings. But just to make you happy, I went and gave you some stars. > I've been acquainted with the theorem since I began teaching myself > tensor calculus at age 14. I would appreciate at least some positive > feedback -- even if only in the form of respect -- for going out of > my way to write an essay for the mathematical community. Surely you agree it's a form of respect to point out your essay is of very little value, from the point of view of academic philosophy or mathematical logic, instead of condescendingly patting you on the head? The web is full of all sorts of essays on the incompleteness theorems. It's obscure why you think your essay in particular should be met with any great excitement, or earn you the respect of the mathematical community. > Do *not* underestimate my intelligence. Your intelligence is not at issue. And while I'm sorry to hear about your suffering, that too is irrelevant to the G�del waffle you've presented. > I've studied this theorem for over ten years and have read the proof > in Goedel's original manuscript. If you have objections or > constructive criticism, state them politely, and I will consider > them. But if you turn our pursuit of truth into a childish peacock > display, I will look elsewhere for discussion, to the detriment > these newsgroups. You will of course look for discussion wherever you like. Peculiar outbursts like this are best avoided. Just ignore any real or perceived insults or peacock displays, and concentrate on the issues at hand. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 29 Sep 2009 12:41 Scott H <zinites_page(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 G�del'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 G�del sentence of a consistent theory is true. There's no Platonism or any philosophising whatever involved. The G�del 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. 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 G�del 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 G�del sentence of Peano arithmetic? Of ZFC? In any case, if the G�del 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.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 29 Sep 2009 13:04 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > There might have been more than one points that I might have > "adamantly insisted" in the past. So unless you spell it out I > couldn't know what we're really talking about here. David's statement, that if T and T* have the same axioms they're the same theory, implies that the language of a theory is determined by its axioms. Surely you recall the fun debate about that from a while back. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Marshall on 29 Sep 2009 10:00 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. Marshall
From: Aatu Koskensilta on 29 Sep 2009 13:13
Marshall <marshall.spight(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? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |