Deconstructing Godel
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From: Nam Nguyen on 10 Feb 2010 21:50 Frederick Williams wrote: > HawkLogic wrote: >> On Feb 10, 8:27 am, Frederick Williams <frederick.willia...(a)tesco.net> >> wrote: >>> HawkLogic wrote: > >>>> Use that. >>>> Set A = >>>> { 1. Both statements in this set are false, >>>> 2.Godelcreated 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. > >> It is the same self-referencing technique that Godel used to prove >> Theorem VI in 1931 (1st Incompleteness >> Theorem), where Flg(k) is the set of axioms and proven formulae. > > Godel's 1931 proof was entirely syntactic, no reference was made to > truth or falsity. > So how did Godel arrive at the conclusion, say, G(T) is *true* but not provable?
From: Frederick Williams on 11 Feb 2010 07:37 Nam Nguyen wrote: > > Frederick Williams wrote: > > HawkLogic wrote: > > > >> It is the same self-referencing technique that Godel used to prove > >> Theorem VI in 1931 (1st Incompleteness > >> Theorem), where Flg(k) is the set of axioms and proven formulae. > > > > Godel's 1931 proof was entirely syntactic, no reference was made to > > truth or falsity. > > > > So how did Godel arrive at the conclusion, say, G(T) is *true* but not provable? I intended to make reference to Godel's Theorem VI [1] the statement and proof of which are entirely syntactic. Since the OP wrote "Theorem VI in 1931" my intention was not unreasonable. [1] 'On formally undecidable propositions of _Principia Mathematica_ and related systems I', pp 596-616 of 'From Frege to G\"odel, a source book in mathematical logic, 1879-1931', ed Jean van Heijenoort, Harvard UP. -- .... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime.
From: Jim Burns on 11 Feb 2010 10:09 Frederick Williams wrote: > MoeBlee wrote: >> On Feb 10, 2:22 pm, Frederick Williams >> <frederick.willia...(a)tesco.net> wrote: >> >>> Yes, FOL _may_ be unsound but who thinks it >> in the least likely? What do you find doubtful in >> the ordinary proof that first order logic is sound? > > Nothing. But I also know that I am fallible. But, what if you're wrong about being fallible? Jim Burns
From: Aatu Koskensilta on 11 Feb 2010 11:14 HawkLogic <hawklogic(a)gmail.com> writes: > Godel seems to have found a way around soundness. This baffling remark may well be the pinnacle of recreational logic, but from the point of view of mathematical logic it's just nonsense. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Feb 2010 11:17
HawkLogic <hawklogic(a)gmail.com> writes: > Godel invented a method which has unexplored consequences. What's the relevance of this vacuous proclamation? Pretty much any mathematical method has "unexplored consequences". This is hardly of any help in making sense of your claim, that G�del may have found a way around soundness. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |