Godel uses peano which is impredicative-thus theorem invalid

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pam69ur
 
Reply Fri 9 May, 2008 07:50 am
the australian philosopher colin leslie dean points out that Godel use
peano axioms which is impredicative-thus his theorem is invalid

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



godel constructs his system P from which he proves his incompleteness
theorem

quote

Godel's first Incompleteness Proof at MROB at MROB



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


axiom 3 of system P reads



Quote:
3. x2(0).x1 ∀ (x2(x1) ⊃ x2(fx1)) ⊃ x1 ∀ (x2(x1))

The principle of mathematical induction: If something is true for x=0, and
if you can show that whenever it is true for y it is also true for y+1,
then it is true for all whole numbers x.

now poincare and others have pointed out the principle of induction is
impredicative
thus invalid -texts books on logic also say impredicative
statements are invalid



quote
Preintuitionism - Wikipedia, the free encyclopedia
peanos 5 th axiom



Quote:
This is the principle of complete induction, it establishes the property
of induction as necessary to the system. Since Peano's axiom is as
infinite as the natural numbers, it is difficult to prove that the
property of P does belong to any x and also x+1. What one can do is say
that, if after some number n of trails that show a property P conserved in
x and x+1, then we may infer that it will still hold to be true after n+1
trails. But this is itself induction. And hence the argument is a vicious
circle.
thus godels theorem is invalid

it does not matter that alot of maths is impredicative just because maths
ueses them does not make them valid- or any proof made useing them
 
Arjen
 
Reply Fri 9 May, 2008 10:32 am
@pam69ur,
Look, the people criticising this theorem missed the entire argument. Peano's axioms clearly show the necessity for an understanding of mathematics on different levels. When one decrales the different levels invalid, offcourse the theorem proves invaled, duh. The thing of it is that there are incomplete "chuncks" in mathematical system. That can be proven by taken a random system (say: ours) and showing the incompleteness in it. Because it exists in one and the reason for it is the existance of different levels it exists in all. That I have not seen counter argued (and with good reason). All this consists in clouding the battlefield so to speak. All I have seen thusfar are attempts at deception; but no proof of the invalidity.
 
 

 
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