Deconstructing Godel
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From: HawkLogic on 9 Feb 2010 18:19 On Feb 9, 12:21 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > HawkLogic wrote: > > > Observations: > > A. There is at least one method (Godel) of generating self-referential > > statements in first-order logic. > > "Traditional" Godelization uses some number theory. In set theory one > can use sets to code syntactic objects. But FOL? Surely that's too > restrictive? > > > [...] > > D. There are informal methods of self-reference that can prove any > > statement (Smullyan, et al). > > Eh? > Godel defined his own system P from the logic of Principia Mathematica and the first-order Peano axioms of arithmetic. Use that. Set A = { { 1. Both statements in this set are false, 2. Godel created a mess. } If 1 is true then both are false, therefore, 1 is not true. If 1 is false then at least one statement is true, therefore 2 is true.
From: HawkLogic on 9 Feb 2010 18:21 On Feb 9, 12:21 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > HawkLogic wrote: > > > Observations: > > A. There is at least one method (Godel) of generating self-referential > > statements in first-order logic. > > "Traditional" Godelization uses some number theory. In set theory one > can use sets to code syntactic objects. But FOL? Surely that's too > restrictive? > > > [...] > > D. There are informal methods of self-reference that can prove any > > statement (Smullyan, et al). > > Eh? Godel defined his own system P from the logic of Principia Mathematica and the first-order Peano axioms of arithmetic. Use that. Set A = { 1. Both statements in this set are false, 2. Godel created a mess. } If 1 is true then both are false, therefore, 1 is not true. If 1 is false then at least one statement is true, therefore 2 is true.
From: Frederick Williams on 10 Feb 2010 09:27 HawkLogic wrote: > > On Feb 9, 12:21 pm, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > HawkLogic wrote: > > > > > Observations: > > > A. There is at least one method (Godel) of generating self-referential > > > statements in first-order logic. > > > > "Traditional" Godelization uses some number theory. In set theory one > > can use sets to code syntactic objects. But FOL? Surely that's too > > restrictive? > > > > > [...] > > > D. There are informal methods of self-reference that can prove any > > > statement (Smullyan, et al). > > > > Eh? > Godel defined his own system P from the logic of Principia Mathematica > and the first-order Peano axioms of arithmetic. > Use that. > > Set A = > { 1. Both statements in this set are false, > 2. Godel created a mess. } > > If 1 is true then both are false, therefore, 1 is not true. > If 1 is false then at least one statement is true, therefore 2 is true. Once is enough :-) You (or Smullyan or someone) are assuming that 1. is a statement S subject to if S is not true then S is false but some (Russell for example) would maintain that 1. is not well-formed and has no truth value. -- .... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime.
From: Aatu Koskensilta on 10 Feb 2010 09:37 HawkLogic <hawklogic(a)gmail.com> writes: > Is there? Is there what? Your observations and conclusions are just vague waffle. In particular, what does the conclusion 2. There are formal methods of self-reference that can prove any statement. mean and how is it supposed to be derived from your peculiar observations? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frederick Williams on 10 Feb 2010 12:20
HawkLogic wrote: > 3. There may be false statements in first-order logic which have been > proven true. May there? Do you have an example? -- .... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime. |