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From: byron on 16 Jun 2010 10:08
t is argued that Godels incompleteness theorems are invalid ie
illegitimate for 5 reasons: he uses the axiom of reducibility- which
is invalid ie illegitimate,he constructs impredicative statement which
is invalid ie illegitimate ,he cant tell us what makes a mathematic
statement true, he falls into two self-contradictions,he ends in three
paradoxes

http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-invalid-illegitimate

First of the two self-contradictions

Godels first theorem ends in paradox ?due to his construction of
impredicative statement
Now the syntactic version of Godels first completeness theorem reads

Proposition VI: To every ?-consistent recursive class c of formulae
there correspond recursive class-signs r, such that neither v Gen r
nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of
r).

But when this is put into plain words we get
http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem

Gödel's first incompleteness theorem states that:
Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete. In particular, for
any consistent, effectively generated formal theory that proves
certain basic arithmetic truths, there is an arithmetical statement
that is true,[1] but not provable in the theory (Kleene 1967, p. 250).

Now truth in mathematics was considered to be if a statement can be
proven then it is true
Ie truth is equated with provability
http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics

??from at least the time of Hilbert's program at the turn of the
twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those
statements which are provable in a formal axiomatic system.
The works of Kurt Gödel, Alan Turing, and others shook this
assumption, with the development of statements that are true but
cannot be proven within the system?

http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem
?Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete. In particular, for
any consistent, effectively generated formal theory that proves
certain basic arithmetic truths, there is an arithmetical statement
that is true,[1] but not provable in the theory (Kleene 1967, p. 250)
For each consistent formal theory T having the required small amount
of number theory
? provability-within-the-theory-T is not the same as truth; the theory
T is incomplete.?

Now it is said godel PROVED
"there are true mathematical statements which cant be proven"
in other words
truth does not equate with proof.

if that theorem is true
then his theorem is false

PROOF
for if the theorem is true
then truth does equate with proof- as he has given proof of a true
statement
but his theorem says
truth does not equate with proof.
thus a paradox
THIS WHAT COMES OF USING IMPREDICATIVE STATEMENTS




GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE

GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE
Now truth in mathematics was considered to be if a statement can be
proven then it is true
Ie truth was s equated with provability
http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics


??from at least the time of Hilbert's program at the turn of the
twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those
statements which are provable in a formal axiomatic system.

The works of Kurt Gödel, Alan Turing, and others shook this
assumption, with the development of statements that are true but
cannot be proven within the system?

Now the syntactic version of Godels first completeness theorem reads
Proposition VI: To every ?-consistent recursive class c of formulae
there correspond recursive class-signs r, such that neither v Gen r
nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of
r).

But when this is put into plain words we get

http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem
?Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete. In particular, for
any consistent, effectively generated formal theory that proves
certain basic arithmetic truths, there is an arithmetical statement
that is true,[1] but not provable in the theory (Kleene 1967, p. 250)

For each consistent formal theory T having the required small amount
of number theory
? provability-within-the-theory-T is not the same as truth; the theory
T is incomplete.?

In other words there are true mathematical statements which cant be
proven
But the fact is Godel cant tell us what makes a mathematical statement
true thus his theorem is meaningless
Ie if Godels theorem said there were gibbly statements that cant be
proven


But if godel cant tell us what a gibbly statement was then we would
say his theorem was meaningless


Now at the time godel wrote his theorem he had no idea of what truth
was as peter smith the Cambridge expert on Godel admitts

http://groups.google.com/group/sci.logi ... 12ee69f0a8

Quote:
Gödel didn't rely on the notion
of truth

but truth is central to his theorem
as peter smith kindly tellls us

http://assets.cambridge.org/97805218...40_excerpt.pdf
Quote:
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,

and godels theorem is

http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir...
Quote:
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.

you see godel referes to true statement
but Gödel didn't rely on the notion
of truth

now because Gödel didn't rely on the notion
of truth he cant tell us what true statements are
thus his theorem is meaningless
From: MoeBlee on 16 Jun 2010 12:34
On Jun 16, 9:08 am, byron <spermato...(a)yahoo.com> wrote:
> t is argued that Godels incompleteness theorems are invalid ie
> illegitimate for 5 reasons: he uses the axiom of reducibility-

For the millionth time: Godel's theorem APPLIES TO certain theories
that have the axiom of reducibility. Godel does not himself rely on
the axiom of reducibility. And where the axiom of reduciblity might be
used in such context, it is to use the axiom WITHIN the system being
discussed thus to draw some conclusions ABOUT the system.

For a simple example (not in Godel's treatment, but just a general
example):

Suppose a system has the axiom "P <-> ~P".

Then we may use that axiom WITHIN the system to prove that the system
proves P and that the system proves ~P, thus to conclude that the
system is inconsistent.

But we did not OURSELVES rely on the axiom "P <-> ~P". We merely
showed that a system that has "P <-> ~P" is inconsistent. That does
NOT entail that we have "P <-> ~P" as an axiom (or principle) in our
own formal (or informal) meta-theory in which we prove things ABOUT a
system that has "P <-> ~P" as an axiom.

In Godel's work he shows that certain systems have a certain property
(among those systems are ones that have the axiom of reducibility and
others that do not have the axiom of reducibility). That Godel proves
things ABOUT systems (some of which have the axiom of reducibility)
does not entail that Godel's own reasonings require that HIS own meta-
theory (informal in his paper, but which we can formalize) adopts the
axiom of reducibility.

That's just for starters. I'll leave the rest of your assertions for
perhaps another day. But at least we should start with this first
misunderstanding of yours

MoeBlee




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