New essay on Goedel's Incompleteness Theorem
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From: Nam Nguyen on 24 Sep 2009 12:40 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> For one example, being our failure to recognize that the encoding of >>> Godel sentence G could be an absolute undecidable, (in the sense of >>> being independent in any extension of Q), as GC could be. >> >> There is no sentence undecidable in all (axiomatisable) extensions of >> Robinson arithmetic. It's obscure what you have in mind. >> > > My wording for the sense of "absolute undecidable" was bad. What I had > in mind is that _assuming_ the consistency of an extension of Q in > question, > it could be *impossible to decide* (hence a sense of "absolute > undecidable") > which one - the formula or its negation - would syntactically contradict > the assumed consistency. > > For instance, T = PA + GC would prove GC for sure but in that case it > could be impossible to know if we still could assume T be consistent. The irony in Godel's work is that he based his proof on the very weakness of Hilbert's program, to attack Hilbert's program! Instead of the accepting the one-size-fit-all arithmetic syntactical formal system, he accepted the one-size-fit-all arithmetic interpretative model. What is the real difference would that make?
From: Nam Nguyen on 24 Sep 2009 13:15 Nam Nguyen wrote: > Nam Nguyen wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> For one example, being our failure to recognize that the encoding of >>>> Godel sentence G could be an absolute undecidable, (in the sense of >>>> being independent in any extension of Q), as GC could be. >>> >>> There is no sentence undecidable in all (axiomatisable) extensions of >>> Robinson arithmetic. It's obscure what you have in mind. >>> >> >> My wording for the sense of "absolute undecidable" was bad. What I had >> in mind is that _assuming_ the consistency of an extension of Q in >> question, >> it could be *impossible to decide* (hence a sense of "absolute >> undecidable") >> which one - the formula or its negation - would syntactically contradict >> the assumed consistency. >> >> For instance, T = PA + GC would prove GC for sure but in that case it >> could be impossible to know if we still could assume T be consistent. > > The irony in Godel's work is that he based his proof on the very > weakness of > Hilbert's program, to attack Hilbert's program! > > Instead of the accepting the one-size-fit-all arithmetic syntactical > formal system, he accepted the one-size-fit-all arithmetic interpretative > model. > > What is the real difference would that make? Seriously. If we accept the one-size-fit-all arithmetic formal system (PM?) there lurks the possibility of an (syntactically) undecidable formula within the system. So why have we chosen to ignore the possibility of a formula whose truth is indeterminable within the one-size-fit-all standard arithmetic model known as the naturals? Of course we could borrow some "higher-priced" principles such as Transfinite Induction, Reflection, ... to prove so-and-so is true or such-and-such is consistent. But, should we keep _borrowing_ the knowledge-money to pay for our knowledge-debt?
From: Nam Nguyen on 24 Sep 2009 14:04 Nam Nguyen wrote: > Nam Nguyen wrote: >> Nam Nguyen wrote: >>> Aatu Koskensilta wrote: >>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>> >>>>> For one example, being our failure to recognize that the encoding of >>>>> Godel sentence G could be an absolute undecidable, (in the sense of >>>>> being independent in any extension of Q), as GC could be. >>>> >>>> There is no sentence undecidable in all (axiomatisable) extensions of >>>> Robinson arithmetic. It's obscure what you have in mind. >>>> >>> >>> My wording for the sense of "absolute undecidable" was bad. What I had >>> in mind is that _assuming_ the consistency of an extension of Q in >>> question, >>> it could be *impossible to decide* (hence a sense of "absolute >>> undecidable") >>> which one - the formula or its negation - would syntactically contradict >>> the assumed consistency. >>> >>> For instance, T = PA + GC would prove GC for sure but in that case it >>> could be impossible to know if we still could assume T be consistent. >> >> The irony in Godel's work is that he based his proof on the very >> weakness of >> Hilbert's program, to attack Hilbert's program! >> >> Instead of the accepting the one-size-fit-all arithmetic syntactical >> formal system, he accepted the one-size-fit-all arithmetic interpretative >> model. >> >> What is the real difference would that make? > > Seriously. If we accept the one-size-fit-all arithmetic formal system > (PM?) there lurks the possibility of an (syntactically) undecidable > formula within the system. So why have we chosen to ignore the possibility > of a formula whose truth is indeterminable within the one-size-fit-all > standard arithmetic model known as the naturals? > > Of course we could borrow some "higher-priced" principles such as > Transfinite > Induction, Reflection, ... to prove so-and-so is true or such-and-such is > consistent. But, should we keep _borrowing_ the knowledge-money to pay for > our knowledge-debt? Along the same line, let me propose the first two Anti-Induction principles, regarding to FOL reasoning. (I) Assuming an extension T of Q be consistent, there exists a theorem of T whose proof we can't know. (II) Assuming an extension T of Q be consistent, there exists a formula whose theorem-hood in T we can't know. We actually can prove (I) as a meta statement. But we have accept (II) as a principle, a thesis.
From: LauLuna on 24 Sep 2009 14:30 On Sep 23, 3:41 am, Scott H <zinites_p...(a)yahoo.com> wrote: > 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. The expression G is about (in the metatheoretical reading of G) cannot contain ANOTHER substitution symbol because it is G itself!
From: LauLuna on 24 Sep 2009 14:44
On Sep 22, 11:17 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > > 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]]]] > . . . This notation suggests there is a free variable in '~Pr S [~Pr S x]' that can be replaced by the Gödel number of '~Pr S x'. But it is not so, The argument in that formula is already the Gödel number of that formula, not a free variable. This makes any chain of reference terminate. You have a formula G that, when metatheoretically interpreted, speaks about the formula G. Full stop. Regards |