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

On Tue, 3 Mar 2009 04:48:59 -0800 (PST), byron <spermatozon@xxxxxxxxx>
wrote:

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

It would be much more accurate to say that this has been _asserted_.
Saying it has been "pointed out" sounds like it's actually true,
which may cause confusion, since of course it's nonsense.

see
http://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=godel+axiom+of+reducibility&source=web&ots=-t22NJE3Mf&sig=idCxcjAEB6yMxY9k3JnKMkmSvhA#PPA119,M1

?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


Also from the link given above
it turns out that godel had no idea what makes a mathematical
statement true as peter smith notes himself

quote
Gödel didn't rely on the notion
of truth

thus his incompletness theorem becomes meaningless


quote

http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir
Gödel's first incompleteness theorem, perhaps the single most
celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but
not provable in the theory, can be constructed.1 That is, any
effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.

And Peter smith notes godel is talking about true mathematical
statements

quote
http://assets.cambridge.org/97805218...40_excerpt.pdf
Godel did is find a general method that enabled him to take any theory
T
strong enough to capture a modest amount of basic arithmetic and
construct a corresponding arithmetical sentence GT which encodes the
claim ?The sentence GT itself is unprovable in theory T?. So G T is
true if and only
if T can?t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,

So with out knowing what makes a mathematical statement true
the incompleteness theorem is meaningless

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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