Godels incompleteness theorem are invalid ie illegitimate

[gimel]

it is argued by colin leslie dean that no matter how faultless godels

logic is Godels incompleteness theorem 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-inva...

 

http://gamahucherpress.yellowgum.com/gamahucher_press_catalogue.htm

 

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

 

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 ω-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© (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%27s_incompleteness_theorems

 

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%27s_incompleteness_theorems

 

“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 ω-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© (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%27s_incompleteness_theorems

 

“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%B6del%27s_incompleteness_theorems

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

[ajb]

A theorem is only as good as "what is put into it". I believe there are systems for which the incompleteness theorems do not apply. In particular the theorems only really make statements about number theory and arithmetic. Not all of mathematics.

 

I am very ignorant of logic and related things, maybe someone else can say more.

[Boberto]

Yes, Godel's proof does assert that truth and proof are not equivalent, however proven theorems are still a subset of true theorems. In other words, all theorems that can be proven are true, but not all true theorems can be proven. Similarly, all squares are rectangles, but not all rectangles are squares.

[vuquta]

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

 

A statement is true if there is an assignment from some given universe that satisifies it.

 

A statement has a very precise definition based on terms and connectives.

 

Here is a statement

 

x² = 2.

 

Here is one with connectives.

 

(x² = 4) ↔ ( x = 2 ) v ( x = -2 )

 

The first statement is true only if your universe contains the irrational numbers.

 

Thus, the set of rational numbers does not satisify the first statement and so that statement is not true using only the rational numbers. Note how this has noithing to do with proof.

 

The second statement is satisfied (true) by a universe that contains 2, -2 4.

 

A proof is a sequence of operations based on the rules of the first order predicate calculus.

 

So, truth depends on the universe you have and a proof is a syntactic sequence of operations that can be performed by a computer.

 

Godels completeness theorems and incompleteness theorems demarcated the two concepts of proof and truth.

[wright496]

it is argued by colin leslie dean that no matter how faultless godels

logic is Godels incompleteness theorem are invalid ie illegitimate

for 5 reasons:

 

 

This is all logic. If his theorem isn't valid then his logic MUST be flawed on some level.

[vuquta]

it is argued by colin leslie dean that no matter how faultless godels

logic is Godels incompleteness theorem are invalid ie illegitimate

for 5 reasons:

 

 

This is all logic. If his theorem isn't valid then his logic MUST be flawed on some level.

 

So, what are these 5 reasons?

 

it is argued by colin leslie dean that no matter how faultless godels

logic is Godels incompleteness theorem are invalid ie illegitimate

for 5 reasons:

 

 

This is all logic. If his theorem isn't valid then his logic MUST be flawed on some level.

 

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

 

Here above is a link to this crackpot.

 

1) "For example Godels uses the axiom of reducibility but this axiom was rejected as being invalid by Russell,"

http://en.wikipedia.org/wiki/Axiom_of_reducibility

"The axiom of reducibility was introduced by Bertrand Russell as part of his ramified theory of types, an attempt to ground mathematics in first-order logic"

 

This axiom is required to avoid the liar's paradox. In ZF set theory, this is implemented in the Axiom schema of replacement:

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

 

Upon further reading, he does not understand that Godel used a meta language for his proof and a godel numbering systen based on Peano arithmatic to code 1st order logic statements satisified by the Natural numbers.

 

Godel was able to prove with Godel numbers, http://en.wikipedia.org/wiki/G%C3%B6del_number that each time you have a set of axioms satisified by the natural numbers, there exists a sentence than cannot be proven to be true by the existing axioms thus far using the first order predicate logic.

 

Whence, the theory of natural numbers cannot be completed using axioms.

 

Tarski then proved his truth theorem/ undefinability theorem to prove that truth in the natural numbers cannot be axiomitized.

This implies it is impossible to provide a recipe book for even something as primitive as the natural numbers.

 

Bell and Machover noted this implies science cannot be axiomitized if the natural numbers cannot be axiomitized since science contains the natural number at least.

 

Now you know the rest of the story.

[khaled]

What if his theorems were based on incompleteness ..?