New essay on Goedel's Incompleteness Theorem
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From: Tim Little on 30 Sep 2009 23:22 On 2009-09-30, Scott H <zinites_page(a)yahoo.com> wrote: > 1. Every statement has a Goedel number. > 2. If a statement is 'about' a number, then its Goedel number is also > 'about' that number. If you like, though the latter is only via its decoding into a statement. I agree that it is reasonable to distinguish between a statement G and its Goedel number G'. The statement G is about the number G', by its construction. By your own principle 2, the number G' is about the same number: G'. Not some next step in "endless reference" G''. - Tim
From: Ross A. Finlayson on 30 Sep 2009 23:43 On Sep 30, 8:22 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2009-09-30, Scott H <zinites_p...(a)yahoo.com> wrote: > > > 1. Every statement has a Goedel number. > > 2. If a statement is 'about' a number, then its Goedel number is also > > 'about' that number. > > If you like, though the latter is only via its decoding into a > statement. I agree that it is reasonable to distinguish between a > statement G and its Goedel number G'. > > The statement G is about the number G', by its construction. By your > own principle 2, the number G' is about the same number: G'. Not some > next step in "endless reference" G''. > > - Tim Oh, the two theories with the same axioms, yes if they're identical axioms then the theories are indeterminate. Is it an axiom of the theory that there exists an identical theory with the same axioms, or that other theories with different axioms are not the same? There are theories where that is not so, yet mutually inconsistent theories. Platonism, relevance, the numbers are natural and uncaring so relevant generally little to relevance of events generally except to the discussion at hand. Theories, theoretic and theoretical language, the model-theoretic discussion generally of theories means that if the theories automatically have the properties that their structural content defines them gives consistent explanations of inconsistent theories. (Rel.) Really it is the bearing to establish the numerical series on the small in the regular generally, quite generally, that I've seen to expand the applicable space of modern proof- and truth-theoretic explanation of the model theoretic, in regards to Platonism. Oh, but that last bit, "in regards to Platonism", why? What if the theory makes the axiom "no other theories"? Obviously regular and the irregular inconsistent theories that are relatively regular generally in correspondence to relevance, sees relevance because theories that are known inconsistent are sometimes sufficient to found trust to make decisions based on inferences of the theory, generally, but also the theory specifically, with acknowledging that the presumed inconsistent theory unfounds determinance. Regards, Ross F.
From: Scott H on 1 Oct 2009 07:17 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. 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.
From: Daryl McCullough on 1 Oct 2009 07:23 Scott H says... > >On Sep 30, 5:23 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> I don't think it makes any sense to say that a Godel number is "about" >> anything. In any case, I don't see where you are getting an infinite >> sequence of G, G', G'', etc. > >I say that a Goedel number is about something when its statement is >about something. That's a very silly thing to say, in my opinion, but be that as it may, how is G'' different from G'? >The infinite sequence arises when the substitution or >'arithmoquine' function is applied repeatedly. That doesn't make any sense. G' is not a sentence with free variables, so it doesn't make any sense to "arithmoquine" it. >I'll be more accurate: G' is about G'', the Goedel number of G', which >G is about. As I told Aatu, I'll leave it open whether G, G', G'', ... >are all basically the same statement. The way you've defined it, only the first one, G, is a statement. The others are numbers. As far as I can understand it, G' = G'' = G'''. Anyway, I don't see how the infinite sequence G, G', G'', etc. in any way helps to explain what's going on in Godel's theorem. It seems to me that all you need to know is the following: 1. Every formula Phi has a corresonding number, #Phi, called the Godel code of Phi. 2. There is a formula Pr in the language of arithmetic such that for every formula Phi, Phi is provable from the axioms of Peano arithmetic <-> Pr(#Phi) is true 3. There is a formula G such that it is provable from the axioms of Peano Arithmetic that G <-> ~Pr(#G) There is no point in introducing an infinite sequence of statements. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 1 Oct 2009 07:29
Scott H says... >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. What is the point of introducing the distinction between G' and G''? You say that G refers to G'. You say that G' refers to the same thing that G does. So G' refers to G'. End of story. 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, but that's *NOT* what was done in Godel's proof. He came up with a single sentence G such that G <-> G is not provable -- Daryl McCullough Ithaca, NY |