Deconstructing Godel
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From: Hawklogic on 11 Feb 2010 12:02 On Feb 11, 10:14 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > HawkLogic <hawklo...(a)gmail.com> writes: > >Godelseems 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.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Self-Reference is a staple of Recreational Logic. Godel made it a staple of Mathematical Logic. He may have brought the nonsense with it.
From: HawkLogic on 11 Feb 2010 12:05 On Feb 11, 10:14 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > HawkLogic <hawklo...(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.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Self-Reference is a staple of Recreational Logic. Godel made it a staple of Mathematical Logic. He may have brought the nonsense with it.
From: HawkLogic on 11 Feb 2010 12:11 On Feb 11, 10:17 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > HawkLogic <hawklo...(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 Gödel may have found a way > around soundness. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus How can soundness and "unexplored consequences" be compatible in any axiomatic system?
From: Aatu Koskensilta on 11 Feb 2010 12:39 HawkLogic <hawklogic(a)gmail.com> writes: > How can soundness and "unexplored consequences" be compatible in any > axiomatic system? This important question has been extensively studied by Erik Freitnautzer, a renowned expert in recreational logic. In his 1967 paper /Unexplored Ordinals/ we find the following result: For sufficiently large (non-constructive but countable) ordinals, the structure of their (non-canonical) tree representation is isomorphic to the (canonical) fiber-bundle on the lattice of Turing degrees of incompatible consistent completions of the set of consequences of the ordinal (when interpreted as a formal theory). (A tree representation in this context corresponds to the metrizable Hilbert space of unexplored consequences of an ordinal. Here we're implicitly relying on Kreisel's conceptual analysis and G�del's interpretation of dialectical materialism.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 11 Feb 2010 12:44
On Feb 11, 11:39 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > HawkLogic <hawklo...(a)gmail.com> writes: > > How can soundness and "unexplored consequences" be compatible in any > > axiomatic system? > > This important question has been extensively studied by Erik > Freitnautzer, a renowned expert in recreational logic. In his 1967 paper > /Unexplored Ordinals/ we find the following result: > > For sufficiently large (non-constructive but countable) ordinals, the > structure of their (non-canonical) tree representation is isomorphic to > the (canonical) fiber-bundle on the lattice of Turing degrees of > incompatible consistent completions of the set of consequences of the > ordinal (when interpreted as a formal theory). > > (A tree representation in this context corresponds to the metrizable > Hilbert space of unexplored consequences of an ordinal. Here we're > implicitly relying on Kreisel's conceptual analysis and Gödel's > interpretation of dialectical materialism.) But wasn't Freitnautzer found guilty of having unnatural relations with a pastrami sandwhich, thus invalidating all his results? MoeBlee |