New essay on Goedel's Incompleteness Theorem
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From: Scott H on 30 Sep 2009 15:43 On Sep 30, 3:20 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Scott H <zinites_p...(a)yahoo.com> writes: > > I know. It was mostly a message to the one-starrer. I don't think he > > understands how much danger he and others are creating. > > This is just silly. Google Groups ratings aren't creating any danger. Fallacy of division. > > At any rate, I have proposed that G refers to its 'reflection' or > > Goedel code, which I have called G' instead of t. > > That you have. You have also completely ignored all comments and > questions regarding your various baffling assertions about such matters. If I had, I wouldn't have responded to them. I even updated my essay based on one of your comments.
From: Daryl McCullough on 30 Sep 2009 16:01 Scott H says... >At any rate, I have proposed that G refers to its 'reflection' or >Goedel code, which I have called G' instead of t. If G' means the Godel code of G, then G' is a natural number, not a statement. So it isn't at all clear what the infinite sequence G, G', G'',... means. If G' is the Godel code of G, then what is G''? -- Daryl McCullough Ithaca, NY
From: Scott H on 30 Sep 2009 16:10 On Sep 30, 4:01 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > If G' means the Godel code of G, then G' is a natural number, > not a statement. So it isn't at all clear what the infinite > sequence G, G', G'',... means. If G' is the Godel code of G, > then what is G''? G'' is the Goedel code of G' expressed in the Goedel code of G'. If statements can be about Goedel numbers, then Goedel numbers can be about Goedel numbers.
From: Scott H on 30 Sep 2009 16:12 On Sep 30, 4:10 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > G'' is the Goedel code of G' expressed in the Goedel code of G'. If it's easier, that should read: expressed in the Goedel code *that is* G'.
From: Daryl McCullough on 30 Sep 2009 16:26
Scott H says... > >On Sep 30, 4:01 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> If G' means the Godel code of G, then G' is a natural number, >> not a statement. So it isn't at all clear what the infinite >> sequence G, G', G'',... means. If G' is the Godel code of G, >> then what is G''? > >G'' is the Goedel code of G' expressed in the Goedel code of G'. If >statements can be about Goedel numbers, then Goedel numbers can be >about Goedel numbers. I can't make any sense of that. Look, to refresh your memory, here are the facts about G: 1. If Phi is any formula, and n is a natural number, then let Phi[n] be the result of replacing the free variables in Phi by the numeral for n. 2. Every formula of arithmetic has an associated Godel code, which is a natural number. If Phi is a formula, then let #Phi be its Godel code. 3. There is a provability predicate Pr for Peano Arithmetic with the property that for any formula Phi, If Phi is provable from the axioms of Peano Arithmetic then Pr(#Phi) is true Conversely, if Phi is not provable from the axioms of Peano Arithmetic, then ~Pr(#Phi) is true. 4. For every formula Phi of arithmetic, there is a corresponding formula G with the property that PA proves G <-> Phi[#G] 5. Applying 4 to the formula ~Pr, we have G <-> ~Pr(#G) So, what is the G', G'', etc. that you are talking about? -- Daryl McCullough Ithaca, NY |