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Science Forum Index » Logic Forum » hermeneutic disproof of godels incompleteness theorem...
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| elsiemelsi... |
 Posted: Mon May 05, 2008 9:27 pm |
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the australian philosopher colin leslie dean gives a hermeneutic disproof
of godels incompleteness theorem
with out any maths he shows godel uses invalid axioms ie axiom of
reducibility
and
invalid statements ie impredicative statements
and points out 3 paradoxes godel falls into
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5 .pdf
quote
Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition
cannot say something about itself , is to drastic... We saw that we can
construct propositions which make statements about themselves,… ((K
Godel , On undecidable propositions of formal mathematical systems in The
undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes,
“it covers ground quite similar to that covered in Godels orgiinal 1931
paper on undecidability,” p.39.)
What Godel understood by "propositions which make statements about
themselves"
is the sense Russell defined them to be
'Whatever involves all of a collection must not be one of the
collection.'
Put otherwise, if to define a collection of objects one must use the
total
collection itself, then the definition is meaningless. This explanation
given by Russell in 1905 was accepted by Poincare' in 1906, who coined
the
term impredicative definition, (Kline's "Mathematics: The Loss of
Certainty"
Note Ponicare called these self referencing statements impredicative
definitions
texts books on logic tell us self referencing ,statements (petitio
principii) are invalid
and
“IV. Every formula derived from the schema
http://www.mrob.com/pub/math/goedel.html
1. (∃u)(v ∀ (u(v) ≡ a))
on substituting for v or u any variables of types n or n + 1 respectively,
and for a a formula which does not contain u free. This axiom represents
the axiom of reducibility (the axiom of comprehension of set theory)”
. Godel uses axiom 1V the axiom of reducibility in his formula 40 where
he states “x is a formula arising from the axiom schema 1V.1 ((K Godel
, On formally undecidable propositions of principia mathematica and
related systems in The undecidable , M, Davis, Raven Press, 1965,p.21
“ [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u
& u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]
x is a formula derived from the axiom-schema IV, 1 by substitution
http://www.mrob.com/pub/math/goedel.html
what godel calls the axiom of reducibility is his streamlined version of
russells axiom
http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps.
"The system P of footnote 48a is Godel’s
streamlined version of Russell’s theory of types built on the natural
numbers as individuals, the system used in [1931]. The last sentence ofthe
footnote allstomindtheotherreferencetosettheoryinthatpaper;
KurtGodel[1931,p. 178] wrote of his comprehension axiom IV, foreshadowing
his approach to set theory, “This axiom plays the role of [Russell’s]
axiom of reducibility (the comprehension axiom of set theory).”
--
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