Re: Godels incompleteness theorem proven invalid
- From: "junegreen25@xxxxxxxxx" <junegreen25@xxxxxxxxx>
- Date: Sun, 9 Nov 2008 22:24:49 -0800 (PST)
On Nov 10, 11:47 am, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:
the australian philosopher colin leslie dean shows Godels incompletenessthe following website to view many pics about shoes, boots, clothes,
theorem is invalid for 7 reasonshttp://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
Godel tells us he uses his system P to prove his incompleteness theorem
quote
In the proof of Proposition VI the only properties of the system P
employed were the following:
1. The class of axioms and the rules of inference (i.e. the relation
"immediate consequence of") are recursively definable (as soon as the
basic signs are replaced in any fashion by natural numbers).
2. Every recursive relation is definable in the system P (in the sense of
Proposition V).
Hence in every formal system that satisfies assumptions 1 and 2 and is
ω-consistent, undecidable propositions exist of the form (x) F(x), where
F is a recursively defined property of natural numbers, and so too in
every extension of such
now system P is is made up of peano axioms and the axioms of PM ie AR
notehttp://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 of
the 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).”
now one reason why his proof is invalid is he uses AR in his proof
1) he uses the axiom of reducibility to make his proof ie his axiom 1v and
in his formular 40
Godel uses 2nd ed of PM but in that ed russell following advice from
wittgensien rejected and abandoned AR ramsey says of AR
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
the standford encyclopdeia of philosophy says of AR
http://plato.stanford.edu/entries/principia-mathematica/
“many critics concluded that the axiom of reducibility was simply too ad
hoc to be justified philosophically”
From Kurt Godels collected works vol 3 p.119
http://books.google.com/books?id=gDzbuUwma5MC&pg=PA119&lpg=PA119&dq=g...
“the axiom of reducibility is generally regarded as the grossest
philosophical expediency “
2)A sceond reason godels theorem is invalid is that he uses impredicative
statements
and text books on logic say such statements are invalid
quote
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
“ 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
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”
Yet Godel uses impredicative definitions in his theorems
“ 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.)
Godel used Peanos axioms but these axioms are impredicative and thus
according to Russell Poincaré and others must be avoided as they lead to
paradox.
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