Deconstructing Godel
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From: Aatu Koskensilta on 11 Feb 2010 12:58 HawkLogic <hawklogic(a)gmail.com> writes: > Self-Reference is a staple of Recreational Logic. Godel made it a > staple of Mathematical Logic. He may have brought the nonsense with > it. Well, I'm afraid I can't make anything of these disconnected proclamations. Admitting defeat I bow out, parting with a helpful parting comment, that should you for some reason care about the actual mathematical content of the incompleteness theorems it might be a good idea to read a text on the subject, such as Torkel Franz�n's _G�del's Theorem_. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Alan Smaill on 11 Feb 2010 12:59 MoeBlee <jazzmobe(a)hotmail.com> writes: > 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 G�del's > > interpretation of dialectical materialism.) > > But wasn't Freitnautzer found guilty of having unnatural relations > with a pastrami sandwhich, thus invalidating all his results? I seem to remember that that was his evil non-standard half-brother of the same name. > MoeBlee > -- Alan Smaill
From: Aatu Koskensilta on 11 Feb 2010 13:02 Alan Smaill <smaill(a)SPAMinf.ed.ac.uk> writes: > MoeBlee <jazzmobe(a)hotmail.com> writes: > >> But wasn't Freitnautzer found guilty of having unnatural relations >> with a pastrami sandwhich, thus invalidating all his results? > > I seem to remember that that was his evil non-standard half-brother > of the same name. MoeBlee's confusion is very understandable. Although Erik Freitnautzer's morally suspect half-brother of the same name was non-standard the two were nevertheless elementarily equivalent. -- 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 15:06 On Feb 11, 11:59 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > 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? > > I seem to remember that that was his evil non-standard half-brother > of the same name. That is not mentioned in MacTruffleboygen's 'Encyclopedia Of Standard and Nonstandard Saints and Sinners'. In fact, as I recall, it is mentioned that not only did Freitnautzer lack siblings but he lacked parents. A curious case, to say the least. Sure, many people have claimed to be parent to Freitnautzer, but all have been discredited by the fact that parentage would violate the para-congruent anti-well founded symmetry of his non-finitary existence. [Note: I spelled 'MacTruffleboygen' with a 'b' rather than with an umlaut over an 'x' as is actually correct but overly pedantic.] MoeBlee
From: MoeBlee on 11 Feb 2010 15:10
On Feb 11, 12:02 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> writes: > > MoeBlee <jazzm...(a)hotmail.com> writes: > > >> But wasn't Freitnautzer found guilty of having unnatural relations > >> with a pastrami sandwhich, thus invalidating all his results? > > > I seem to remember that that was his evil non-standard half-brother > > of the same name. > > MoeBlee's confusion is very understandable. Although Erik Freitnautzer's > morally suspect half-brother of the same name was non-standard the two > were nevertheless elementarily equivalent. Well, given the non-existence of his half-brother, that follows vacuously. Though it is still an open question whether the non- existence of his elementary equivalent half-brother implies the non- existence of Freitnautzer himself. MoeBlee |