New essay on Goedel's Incompleteness Theorem
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From: Scott H on 22 Sep 2009 21:42 On Sep 22, 5:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Are you saying G is infinite in length? No, but I am saying that the substitution process can be carried out infinitely.
From: Nam Nguyen on 22 Sep 2009 22:18 Scott H wrote: > On Sep 22, 5:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Are you saying G is infinite in length? > > No, but I am saying that the substitution process can be carried out > infinitely. What you said: >> In symbols, >> >> G = ~ Pr S [~ Pr S x] >> = ~ Pr [~ Pr S [~ Pr S x]] >> = ~ Pr [~ Pr [~ Pr S [~ Pr S x]]] >> = ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]] >> . . . Note the phrase "In _symbols_" you used and note that G is always on the left side of "=". If that doesn't mean G is _syntactically_ infinite, I'm not sure what you were saying!
From: Daryl McCullough on 23 Sep 2009 08:17 Scott H says... >The infinite sequence, I argue, arises when we consider that >statements G'', G''', arise from the repeated application of the >substitution function used in G's construction. Why would you repeatedly apply the substitution function? >This repeated application results in an endless semantic reference. > >In symbols, > >G = ~ Pr S [~ Pr S x] I'm not sure about your notation here, but if we pretend that there is a built-in S(x) with the property that if n is the Godel number of a sentence Phi, then S(n) is the Godel number of the sentence Phi' obtained from Phi by substituting all occurrences of free variables by the numeral for n. In that case, we can let G_0 be the formula ~Pr(S(x)) and we can let #G_0 be the Godel number of G_0, then we can define G to be the formula ~Pr(S(#G_0)) I think that's what you mean. >= ~ Pr [~ Pr S [~ Pr S x]] That step is not legitimate. G is not equal to that expression. It is provably equivalent to that expression, but it is not equal. >= ~ Pr [~ Pr [~ Pr S [~ Pr S x]]] >= ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]] >. . . This infinite sequence doesn't really have any significance. You can certainly come up with an infinite sequence of formulas G, G', G'', G''', etc that are all provably equivalent, but there is no significance to this sequence that I can see. What I think you are doing is performing the "rewrite rule" S(~Pr(S(x)) --> ~Pr(S(~Pr(S(x)))) But you can't *KEEP* performing the rewrite rule on the innermost "S", because that S is being *quoted*. It's a String. The fact that two different strings denote the same object does not mean that you can substitute one for the other in an "opaque" context. If John says "The saboteur planted the bomb", and the saboteur happens to be Frank, then that does *NOT* mean that John said "Frank planted the bomb". Even though "Frank" and "the saboteur" have the same denotation, you can't substitute one for the other inside a quotation. Provability is opaque in this sense. You can't substitute equals for equals inside the provability predicate. So your infinite sequence is *NOT* legitimate. -- Daryl McCullough Ithaca, NY
From: Scott H on 23 Sep 2009 17:21 On Sep 23, 8:17 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Scott H says... > >= ~ Pr [~ Pr S [~ Pr S x]] > > That step is not legitimate. G is not equal to that > expression. It is provably equivalent to that expression, > but it is not equal. All right; my mistake. I should have used the equivalence symbol. > You can certainly come up with > an infinite sequence of formulas > > G, G', G'', G''', etc > > that are all provably equivalent, but there is > no significance to this sequence that I can see. There is, in my mind, *some* significance to the idea of endless reference as distinct from self-reference, as I point out near the end of the third section. For example, while "This statement is false" may be paradoxical, the truth values of The following is false: The following is false: The following is false: ... might still be well-defined. It is my aim in the section on supernatural numbers to offer an intuitive description of PA + ~G (or ZFC + ~G, your choice), and to do this, I required the idea of endless reference.
From: Aatu Koskensilta on 23 Sep 2009 17:46
Scott H <zinites_page(a)yahoo.com> writes: > It is my aim in the section on supernatural numbers to offer an > intuitive description of PA + ~G (or ZFC + ~G, your choice), and to do > this, I required the idea of endless reference. Alas, your intuitive descriptions are hopelessly vague and for most part entirely unconnected to anything in the actual technical content of the theorems and their proofs. This is true in particular of the passage I quoted earlier We must remember, however, that G�del's theorem is founded not on self-reference but 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'. as well as such passages as 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. As said, I can't really fathom the exact nature of your confusion -- taken literally much of what you write is simply nonsense, and perhaps owing to my pedantic nature, I'm unable to come up with a charitable interpretation -- so unfortunately my comments are restricted to somewhat boring generalities. No deep understanding of your thought processes is necessary, however, for the assessment that whatever line of thought led you to the ideas expressed in these passages doesn't have much to do with anything we find in the theorems and their proofs, or the various erudite and less erudite matters that come up in connexion to them in (the competently written part of) the literature. This alone should give you pause. Your essay is the sort of excited stuff intelligent people often come up upon learning of the incompleteness theorems. Heed the following wise words from the renowned French eccentric and proof-theorist Jean-Yves Girard: The usual experience of the proof-theorist with G�del's theorems is that, in a first step one gets so struck one tries to reformulate one's personal view of the world to fit the contents of the theorems. Later on, with reasonable experience of these results, this kind of "dramatic" consequences appear as ridiculous extrapolations. The situation is quite different with outsiders [ - - - ] Now, lest I sound too negative, I'll add that in my experience intelligent people who make it their business to thoroughly understand the mathematics involved will eventually see the light, and, armed with a sober understanding of the subject, come to view their earlier bizarre tirades, such as http://groups.google.com/group/comp.ai.philosophy/msg/b27dfd040b49b45b with nostalgic amusement. Two books by the late G�del police Torkel Franz�n, _G�del's Theorem_ and _Inexhaustibility_ are excellent sources for such an understanding. That said, there is of course no pressing reason for anyone to take any interest in the incompleteness theorems, or this or that technical result in logic -- and unless one is willing to put in the effort and actually study the mathematical, conceptual and philosophical issues involved, in quite some detail, it is pointless to offer general reflections and musings as to the significance or interpretation of such results. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |