5 reasons why Godels incompleteness theorem invalid

The Autralian philosopher Colin Leslie Dean points out
Godels theorems are invalid for 5 reasons: he uses the axiom of
reducibility- which is invalid, he uses the axiom of choice, he
constructs impredicative statements - which are invalid ,he miss uses
the theory of types, he falls into 3 paradoxes

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf


AXIOM OF REDUCIBILITY
(1) Godel uses the axiom of reducibility axiom 1V of his system is the
axiom of reducibility â??As Godel says â??this axiom represents the axiom
of reducibility (comprehension axiom of set theory)â?? (K Godel , On
formally undecidable propositions of principia mathematica and related
systems in The undecidable , M, Davis, Raven Press, 1965,p.12-13)
.. Godel uses axiom 1V the axiom of reducibility in his formula 40 where
he states â??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

â?? [40. R-Ax(x) â?¡ (â??u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u
& u Fr y & Form(y) & x = u â??x {v Gen [[R(u)*E(R(v))] Aeq y]}]

x is a formula derived from the axiom-schema IV, 1 by substitution â??
http://www.mrob.com/pub/math/goedel.html


( 2) â??As a corollary, the axiom of reducibility was banished as
irrelevant to mathematics ... The axiom has been regarded as re-instating
the semantic paradoxesâ?? -
http://mind.oxfordjournals.org/cgi/reprint/107/428/823.pdf
2)â??does this mean the paradoxes are reinstated. The answer seems to be
yes and noâ?? - http://fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf
)

3) It has been repeatedly pointed out this Axiom obliterates the
distinction according to levels and compromises the vicious-circle
principle in the very specific form stated by Russell. But The
philosopher
and logician FrankRamsey (1903-1930) was the first to notice that the
axiom of reducibility in effect collapses the hierarchy of levels, so
that
the hierarchy is entirely superfluous in presence of the axiom.
(http://www.helsinki.fi/filosofia/gts/ramsay.pdf)

4) Russell Ramsey and Wittgenstein regarded it as illegitimate Russell
abandoned this axiom and many believe it is illegitimate and must be not
used in mathematics

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY


IMPREDICATIVE DEFINITIONS
Godel used impredicative definitions

Ponicare Russell and philosophers argue these types of definitions are
invalid Ponicare Russell point out that they lead to contradictions in
mathematics

Quote from Godel
â?? 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,â?¦


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

Godels has argued that impredicative definitions destroy mathematics and
make it false

http://www.friesian.com/goedel/chap-1.htm

Gödel has offered a rather complex analysis of the vicious circle
principle and its devastating effects on classical mathematics
culminating
in the conclusion that because it "destroys the derivation of mathematics
from logic, effected by Dedekind and Frege, and a good deal of modern
mathematics itself" he would "consider this rather as a proof that the
vicious circle principle is false than that classical mathematics is
false

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