Re: Godels use of the axiom of reducibility makes his incompleteness theorem invalid

On Mar 4, 6:20 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Mar 4, 11:33 am, byron <spermato...@xxxxxxxxx> wrote:



It has been pointed out on maths forums that Godels use of the axiom
of reducibility makes his incompleteness theorem invalid

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

the university of california maths depart notes godels system p uses
AR

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 of the
footnote
callstomindtheotherreferencetosettheoryinthatpaper; 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
The stanford encyclopedia of philosophy ststes

http://plato.stanford.edu/entries/principia-mathematica/

many critics concluded that the axiom of reducibility was simply too
ad hoc to be justified philosophically.

Ramsay states like wise about AR

its introduction into mathematics is inexusable.

Such an axiom has no place in mathematics,
. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

Even the editors of godels collected works note

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 “

Now as i showed godels uses AR in his system P to prove his
incompleteness theorem
but
as we also see that axiom is not regarded as valid
thus by useing an invalid axiom godels incompleteness theorem must
like be invalid-even if others have proven by other means an
incompleteness theorem
it remains that WHAT GODEL DID is invalid

Now it has been pointed out that

The systems to which we apply Godel's theorem nowadays
DON'T EVEN NEED an "axiom of reducibility

but my point is
It is what godel did and what he did was use AR -thus making HIS
incompleteness theorem invalid

But Gödel did not use the system P, he proved a result about it in a
much weaker metatheory which does not have the axiom of reducibility.
There is nothing wrong with the theorem, regardless of what you think
about the axiom of reducibility. Furthermore the result is true of
many other systems which do not have the axiom of reducibility.



you say

But Gödel did not use the system P, he proved a result about it in a
much weaker metatheory which does not have the axiom of reducibility.


in godels original paper godels tells us he uses system P to prove his
incompleteness theorem

note he starts by saying

"In the proof of Proposition VI "

then he says

"the only properties of the system P
employed were the following:"

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).

and as the uni of california notes in system P is AR
.

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