Deconstructing Godel
in [Logic]
|
Prev: I LOVE YOU QUOTES & LOVE METER
Next: geometry precisely defining ellipsis and how infinity is in the midsection #426 Correcting Math
From: HawkLogic on 9 Feb 2010 12:56 Observations: A. There is at least one method (Godel) of generating self-referential statements in first-order logic. B. There is no evidence that there are not alternative methods of self- reference. C. There are no guidelines to recognize the use of alternative (non- Godelian) methods of self-reference. D. There are informal methods of self-reference that can prove any statement (Smullyan, et al). E. Godel formalized the use of self-reference in logical proofs. Conclusions: 1. Some proofs may use unknown methods of self-reference. 2. There are formal methods of self-reference that can prove any statement. 3. There may be false statements in first-order logic which have been proven true. 4. There is no way to know. Is there?
From: Frederick Williams on 9 Feb 2010 13:21 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? -- .... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime.
From: HawkLogic on 9 Feb 2010 18:10 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. An example from Smullyan: ___________________________________ | | | 1. Both statements in this box 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 18:14 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. An example from Smullyan: ___________________________________ | 1. Both statements in this box 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 18:18
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. |