New essay on Goedel's Incompleteness Theorem
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From: Scott H on 22 Sep 2009 17:17 On Sep 22, 9:41 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I'm not sure why you say that it is an "infinite" substitution. > There are two sentences involved in Godel's proof. First, > we construct the sentence G. Then we prove (using the construction) > that > > G <-> not Provable(#G) > > where #G means the Godel code of G. If you want to give > not Provable(#G) a new name, G', then you have two > sentences: > > G <-> G' > > I don't see that there is an infinite sequence of sentences. 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. This repeated application results in an endless semantic reference. 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]]]] .. . . It's a bit like looking up the meaning of a word in a dictionary, and having to look up another word, and another word, and so on, to infinity.
From: Newberry on 22 Sep 2009 17:43 On Sep 22, 2:17 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > On Sep 22, 9:41 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: > > > I'm not sure why you say that it is an "infinite" substitution. > > There are two sentences involved in Godel's proof. First, > > we construct the sentence G. Then we prove (using the construction) > > that > > > G <-> not Provable(#G) > > > where #G means the Godel code of G. If you want to give > > not Provable(#G) a new name, G', then you have two > > sentences: > > > G <-> G' > > > I don't see that there is an infinite sequence of sentences. > > 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. This repeated > application results in an endless semantic reference. > > 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]]]] > . . . > > It's a bit like looking up the meaning of a word in a dictionary, and > having to look up another word, and another word, and so on, to > infinity. Is it a litlle bit like This sentence is not true "This sentence is not true" is not true ' "This sentence is not true" is not true' is not true ?
From: Nam Nguyen on 22 Sep 2009 17:46 Scott H wrote: > On Sep 22, 9:41 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: >> I'm not sure why you say that it is an "infinite" substitution. >> There are two sentences involved in Godel's proof. First, >> we construct the sentence G. Then we prove (using the construction) >> that >> >> G <-> not Provable(#G) >> >> where #G means the Godel code of G. If you want to give >> not Provable(#G) a new name, G', then you have two >> sentences: >> >> G <-> G' >> >> I don't see that there is an infinite sequence of sentences. > > 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. This repeated > application results in an endless semantic reference. > > 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]]]] > . . . > > It's a bit like looking up the meaning of a word in a dictionary, and > having to look up another word, and another word, and so on, to > infinity. Are you saying G is infinite in length?
From: LauLuna on 22 Sep 2009 18:47 On Sep 22, 4:03 am, Scott H <zinites_p...(a)yahoo.com> wrote: > The substitution or arithmoquine function, which replaces the free > variable of a property with the symbol for that property, allows an > infinite substitution process to take place. Ultimately, we obtain a > statement similar to: > > The following is unprovable: The following is unprovable: The > following is unprovable: ... No. That's wrong. The Gödel sentence G is not about the Gödel number of a property (i.e. a formula with one free variable) but about the Gödel number of a sentence -a formula with no free variables- namely G. Let me speak in terms of diagonalization rather than arithmoquinization; I'm more familiar with the former. You have a predicate with just one free variable P(x). You substitute the Gödel number of P(x) -say b- for x in P(x) and obtain P(b). P(b) is the diagonalization of P(x). Let d be its Gödel number. Let diag(b) = d. Consider the open formula P(diag(x)) and let its Gödel number be k. Then P(diag(k)) is about diag(k), that is, the Gödel number of the result of substituting k for x in the formula whose Gödel number is k -the formula P(diag(x))-, a result which is precisely P(diag(k)). So, P(diag(k)) can be read as speaking about its own Gödel number diag (k). If P(x) is the provability predicate, then P(diag(k)) is the Gödel sentence G. But note that G contains neither free variables nor the Gödel number of a formula with free variables. Its argument is not the Gödel number of some open formula but its own Gödel number. There is no further replacement of variables because there are no free variables left in G and no Gödel number of a formula with free variables. k is the Gödel number of P(diag(x)), which has a free variable, but G does not contain the numeral for k but for diag(k). The circle is, so to say, completed. Regards
From: Scott H on 22 Sep 2009 21:41
On Sep 22, 6:47 pm, LauLuna <laureanol...(a)yahoo.es> wrote: > There is no further replacement of variables because there > are no free variables left in G and no Gödel number of a > formula with free variables. That's true; there are no free variables left in G after substitution. However, the expression G is *about* contains another substitution symbol, and can itself be replaced. This process can be carried out ad infinitum, and we do, in fact, arrive at endless propositional reference. |