New essay on Goedel's Incompleteness Theorem
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From: David C. Ullrich on 1 Oct 2009 07:56 On Thu, 1 Oct 2009 04:17:50 -0700 (PDT), Scott H <zinites_page(a)yahoo.com> wrote: >I have already warned the one-starrer that I have been retching in >anguish for three years and that I have *deliberately avoided suicide* >to study Goedel's Incompleteness Theorem. He does not seem to realize >or care about the danger of his actions. If he continues to hide and >one-stars this post, we will take it as further evidence of his lack >of empathy and his willingness to 'cross a line' with someone on the >brink of suicide. This, in turn, will reflect on the moral character >of the entire country. You need to talk to a doctor about all this, perhaps several doctors. Seriously. See, the world is the way the world is, and people are the way they are. If one person clicking one button on Google Groups makes you worry about suicide that's something you need to fix. No, that's _not_ fair. I'm not talking about the way things should be, I'm talking about the way things _are_. You're not going to ever get _everyone_ to approve of anything you do - if your survival depends on universal approbation you simply need to change that. >Remember, one-starrer: I wrote this essay for *you*. > >On Sep 30, 11:12 pm, Tim Little <t...(a)little-possums.net> wrote: >> On 2009-09-30, Scott H <zinites_p...(a)yahoo.com> wrote: >> >> > At any rate, I have proposed that G refers to its 'reflection' or >> > Goedel code, which I have called G' instead of t. >> >> Yes, statement G refers to the number G'. G' does not literally refer >> to anything, as it is not a statement. If it is interpreted as a >> statement via the decoding, that statement is G and refers to G'. >> There is no G'', and no endless reference. > >I've deliberately left it an open question whether G, G', G'', ... are >the same statement. Calling the referent of G' G'' does not mean that >G' and G'' are not equal. > >It is important to consider Goedel's theorem from the perspective of >endless reference because a self-referential statement and its >analogous endlessly referential statement may have different >properties. For instance, "This statement is false," we think as >paradoxical, as opposed to > >The following is false: The following is false: The following is >false: ... > >which may have a truth value of T or F, as actual self-reference is >avoided. Knowing this, how would you prove that G, G', G'' ... were >really the same? > >Come to think of it, this is what sci.math may be looking for: an >equivalent of > >~Pr S[~Pr S x] >~Pr [~Pr S[~Pr S x]] >~Pr [~Pr [~Pr S[~Pr S x]]] >. . . > >written as > >G_0 >G_1 --> G'_0 >G_2 --> G'_1 --> G''_0 >. . . > >I have chosen to write G -> G' -> G'' -> ... because G_0, G_1, >G_2, ... are all equivalent. I think this will simplify the essay; >however, as always, I'm open to constructive feedback. David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
From: Jesse F. Hughes on 1 Oct 2009 08:01 Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >> Shoenfield does *not* claim L(T) is determined by its axioms. Here's >> the quote, once again: >> >> "We consider a language to be completely specified when its symbols >> and formulas are specified. This makes a language a purely >> syntactical object. Of course, most of our languages will have a >> meaning (or several meanings); but the meaning is not considered to >> be part of the language. We shall designate the language of a formal >> system F by L(F)." >> >> "The next part of a formal system consists of its /axioms/. Our >> only requirement on these is that each axiom shall be a formula of >> the language of the formal system." [p. 4 in my copy of Shoenfield] >> >> The first part of a formal system is the language. The *next* part is >> the axioms. Clearly, then, the axioms do not determine the language. > > You sounded like a kindergarten kid reading technical book for the first > time, reading _too literally_ what's in the book! Where in Shoenfield's > book did he say that if one doesn't consider language as the first > part of formal system, one's reasoning would fall apart? Speaking of misreading, where did *I* say that if one doesn't consider language as the first part of formal system, one's reasoning would fall apart? Anyway, to hell (once again) with this pointless "debate". As you can see, Shoenfield wrote clearly and contradicts your interpretation. -- Jesse F. Hughes "Even I, who know beyond doubt that my death will be caused by a silly girl, will not hesitate when that girl passes by." -- Merlin, as reported by John Steinbeck.
From: Scott H on 1 Oct 2009 08:36 I know we're getting off the subject, David C. Ullrich, so I'm going to be brief. On Oct 1, 7:56 am, David C. Ullrich <dullr...(a)sprynet.com> wrote: > You need to talk to a doctor about all this, perhaps several doctors. > > Seriously. I have already talked to many doctors and none of their treatments have ever worked. Not only that, but they have written false statements about me in their reports that appeared to indicate a possible profit motive. > See, the world is the way the world is, and people are the way they > are. Vacuous statement likely intended to inspire submission to torture. > If one person clicking one button on Google Groups makes > you worry about suicide that's something you need to fix. To infer that would be an instance of the fallacy of the single cause. > No, that's _not_ fair. I'm not talking about the way things > should be, I'm talking about the way things _are_. You're > not going to ever get _everyone_ to approve of anything > you do - if your survival depends on universal approbation > you simply need to change that. To accuse me of seeking approval on the first draft of my essay would be a straw man. In closing, I will make one remark: My feelings matter. I'll leave you to scratch your head figuring out what that means.
From: Aatu Koskensilta on 1 Oct 2009 08:42 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > I think perhaps what you mean is for there to be a sequence of > different formulas G, G', G'', ... such that > > G == G' is not provable > G' == G'' is not provable > G'' == G''' is not provable > etc. > > It's possible that you could construct such a sequence, [ - - -] We can construct such a sequence using the recursion theorem: let f be a recursive function such that f(x) = [~Prov_T({e}(x+1))] where the square brackets take a formula to its G�del number, {} are the Kleene brackets, and e is an index of f. For no n is f(n) the code for a provable sentence. I don't think we can do without recursion theoretic machinations -- in particular, the usual devices of provability logic appear to be insufficient. I could be wrong. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 1 Oct 2009 08:47
Scott H <zinites_page(a)yahoo.com> writes: > I have already warned the one-starrer that I have been retching in > anguish for three years and that I have *deliberately avoided suicide* > to study Goedel's Incompleteness Theorem. He does not seem to realize > or care about the danger of his actions. If he continues to hide and > one-stars this post, we will take it as further evidence of his lack > of empathy and his willingness to 'cross a line' with someone on the > brink of suicide. This, in turn, will reflect on the moral character > of the entire country. Which country? In any case, it's futile and pointless to obsess over Google Groups ratings. There's nothing you or anyone can do about such matters. I second David's suggestion you seek whatever help you need -- and if news squabbles and Google Groups ratings give you such anguish perhaps you should consider quitting news altogether. What you describe sounds very serious. > I've deliberately left it an open question whether G, G', G'', ... are > the same statement. This makes about as much sense as leaving it an open question whether addition is associative. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |