Deconstructing Godel
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From: HawkLogic on 10 Feb 2010 14:02 On Feb 10, 8:27 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > 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? > >Godeldefined 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.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. > > -- > ... A lamprophyre containing small phenocrysts of olivine and > augite, and usually also biotite or an amphibole, in a glassy > groundmass containing analcime. 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.
From: HawkLogic on 10 Feb 2010 14:03 On Feb 10, 8:37 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > HawkLogic <hawklo...(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.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Is there any way to know that accepted methods of logical proof do not lead to contradiction. The observations I made simple and straight forward. Could you specify which one(s) you do not understand. There are recreational methods of proof for which Godel invented a way to make them mathematical. The (peculiar?) observations are fundamental to the substance of recreational logic and mathematics
From: HawkLogic on 10 Feb 2010 14:04 On Feb 10, 11:20 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > 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. No, the point being that proof methods may have created one without warning.
From: MoeBlee on 10 Feb 2010 15:16 On Feb 10, 1:03 pm, HawkLogic <hawklo...(a)gmail.com> wrote: > Is there any way to know that accepted methods of logical proof do not > lead to contradiction. Yes.You're not familiar with the soundness theorem for first order logic? MoeBlee
From: MoeBlee on 10 Feb 2010 15:17
On Feb 10, 2:16 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Feb 10, 1:03 pm, HawkLogic <hawklo...(a)gmail.com> wrote: > > > Is there any way to know that accepted methods of logical proof do not > > lead to contradiction. > > Yes.You're not familiar with the soundness theorem for first order > logic? Anyway, what specific logical principles do you doubt preserve consistency? MoeBlee |