[sim (OP)]

Godels theorem is invalid as his  G statement is banned by an axiom of the system he uses to prove his theorem

http://gamahucherpress.yellowgum.com/wp-content/uploads/GODEL5.pdf

a flaw in  theorem Godels sentence G is outlawed by the very axiom he uses to prove his theorem
ie the axiom of reducibiilty  AR -thus his proof is invalid

http://www.enotes.com/topic/Axiom_of_reducibility

russells axiom of reducibility was formed such that impredicative statements were banned


but godels uses this AR axiom in his incompleteness proof ie axiom 1v
and formular 40

and as godel states he is useing the logic of PM ie AR

P is essentially the system obtained by superimposing on the Peano axioms the logic of PM ie AR axiom of reducibility

now godel constructs an impredicative statement G which AR was meant
to ban

The impredicative statement Godel constructs is
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#F...

the corresponding Gödel sentence G asserts: G cannot be proved to be true within the theory T

now godels use of AR bans godels G statement

thus godel cannot then go on to give a proof by useing a statement his own axiom bans
but in doing so he invalidates his whole proof

[David Cooper]

Gödel uses an illegal step in his incompleteness theorem, carelessly throwing away an infinite recursion during the diagonalisation process under the mistaken assumption that this doesn't matter. This can be shown more simply though by looking at the glaring faults in the simpler cases which led to him coming up with his theorem.

"This statement is true." That is not in reality a true statement at all, but merely holds the illusion of being true. It too contains an infinite recursion which leads to it failing to compute. This is much easier to see after you've examined the following case:-

"The next statement is true. The previous statement is true." Here, to test the truth of the first sentence, you have to test the truth of the second, and to test the truth of the second you have to test the truth of the first. This process to assess the truth of either statement is infinite and never computes. Gödel made the mistake of thinking that such cases do compute based on the superficial illusion of them being both meaningful and true, but they are actually empty.

How do we test the truth of "the thing in the box is a cat"? We open the box to see if it's a cat, and if it's a cat, we determine that the sentence is true: we substitute "the thing in the box" with whatever is found to be in the box, and when we do that we end up with "a cat is a cat", and that is clearly true.

How do we test the truth of "this box is true"? We look at the box and ask if it can be true or false. It can't be either, so in is inappropriate to apply truth values to the statement. (In computer programming, it is common to assert that statements that look like this one can be true, but that is due to them being written in shorthand - in such cases, there is an unstated verb involved, and it's the statement with the verb included that's actually being tested for truth. In a case where there is no verb, there can be no truth status to test for.)

With "this statement is true", how should be test it? We should do so by substituting "this statement" with whatever it represents, and in this case that's the entire statement. So, we then have `"this statement is true" is true'. We then have to test that statement to see if it is true, and again we do it through substitution: "`"this statement is true" is true' is true". This process of substitutions goes on infinitely without ever resolving the issue. It does not compute.

This is also why "this statement is a lie" isn't a paradox - there is nothing there to attach a truth value to in any meaningful way because again there is infinite recursion.

Gödel's "this statement is undecided" is just as invalid.

I've been told that there are alternative incompleteness theorems which cover the same ground in different ways, so the removal of Gödel's incompleteness theorem doesn't necessarily change anything for mathematics, but I haven't looked at them to see if they stand up either. AGI will look at all this some day and settle the matter definitively, so I don't intend to put any of my time into direct exploration of anything that might be bogus mathematics - there are more efficient ways of getting to the same end point. What is certain though is that Gödel's incompleteness theorem contains a fundamental error and that it does not stand up to scrutiny.

[evan_au]

Godels theorem is invalid

Godel's theorem may be loosely paraphrased as follows: "Godel showed mathematically that you can't prove everything mathematically"...

As commented, this sounds like an infinite recursion.

The difference is that Godel only needed to provide one example to show that his theorem is true.

To prove Godel wrong, you need to show that every possible contradiction-free mathematical system is complete, which is a bigger infinite regress than the one of which you criticise Godel.

Philosophically, I see Godel's theorem as just another step in cutting down the hubris of humanity.

See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

[mad aetherist]

Godels theorem is invalid

Godel's theorem may be loosely paraphrased as follows: "Godel showed mathematically that you can't prove everything mathematically"... As commented, this sounds like an infinite recursion. The difference is that Godel only needed to provide one example to show that his theorem is true.
To prove Godel wrong, you need to show that every possible contradiction-free mathematical system is complete, which is a bigger infinite regress than the one of which you criticise Godel. Philosophically, I see Godel's theorem as just another step in cutting down the hubris of humanity.
See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Surely math is just a language. Language boils down to alphabet & words.  Words come from a dictionary, & words depend on other words for definition. Therefore this is just one more way to say that all arguments are circular. There are other reasons why all arguments are circular.  Anything circular cannot be a proof. But the bigger the circle the better the "proof".

[David Cooper]

Godel's theorem may be loosely paraphrased as follows: "Godel showed mathematically that you can't prove everything mathematically"...

As commented, this sounds like an infinite recursion.

The difference is that Godel only needed to provide one example to show that his theorem is true.

But he failed to provide any example. All he did was provide a faulty example which contained an error based on the idea that "this statement is true" is true. That was an error in the foundations which he then built upon.

To prove Godel wrong, you need to show that every possible contradiction-free mathematical system is complete, which is a bigger infinite regress than the one of which you criticise Godel.

To prove his theorem wrong, you only have to prove his theorem wrong. If by luck his conclusion is valid and can be proved by other means, that doesn't make his faulty theorem valid.