Proof Gödels incompleteness theorem is invalid

[das]

The Australian philosopher Colin leslie Dean points Godels theorem is

invald for 5 reasons: he uses the axiom of reducibility, he uses the

axiom of choice, he miss uses the theory of types, he ends in 3 paradoxes and he uses vicious circle statements. Dean out that Godels uses

impredicative statements in his incomplertenes theorem that makes it

invalid

 

 

 

 

GÖDEL’S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS

GÖDEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS

CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS

By

COLIN LESLIE DEAN

B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,

M.A (PSYCHOANALYTIC STUDIES), MASTER OF PSYCHOANALYTIC STUDIES, GRAD

CERT

(LITERARY STUDIES)

GAMAHUCHER PRESS WEST GEELONG, VICTORIA AUSTRALIA

2007

 

 

Quote from Godel

“ The solution suggested by Whitehead and Russell, that a proposition

cannot say something about itself , is to drastic... We saw that we can

construct propositions which make statements about themselves,… ((K Godel

, On undecidable propositions of formal mathematical systems in The

undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes,

“it covers ground quite similar to that covered in Godels orgiinal 1931

paper on undecidability,” p.39.)

 

 

What Godel understood by "propositions which make statements about

themselves"

 

is the sense Russell defined them to be

 

'Whatever involves all of a collection must not be one of the

collection.'Put otherwise, if to define a collection of objects one must

use the total

collection itself, then the definition is meaningless. This explanation

given by Russell in 1905 was accepted by Poincare' in 1906, who coined

the

term impredicative definition, (Kline's "Mathematics: The Loss of

Certainty"

 

Note Ponicare called these self referencing statements impredicative

definitions

 

texts books on logic tell us self referencing ,statements (petitio

principii) are invalid

 

Godels has argued that impredicative definitions destroy mathematics and

make it false

 

 

 

Gödel has offered a rather complex analysis of the vicious circle

principle and its devastating effects on classical mathematics

culminating

in the conclusion that because it "destroys the derivation of mathematics

from logic, effected by Dedekind and Frege, and a good deal of modern

mathematics itself" he would "consider this rather as a proof that the

vicious circle principle is false than that classical mathematics is

false”

 

AXIOM OF REDUCIBILITY

(1) Godel uses the axiom of reducibility axiom 1V of his system is the axiom of reducibility “As Godel says “this axiom represents the axiom of reducibility (comprehension axiom of set theory)” (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.12-13. Godel uses axiom 1V the axiom of reducibility in his formula 40 where he states “x is a formula arising from the axiom schema 1V.1 ((K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.21

( 2) “As a corollary, the axiom of reducibility was banished as irrelevant to mathematics ... The axiom has been regarded as re-instating the semantic paradoxes”

2)“does this mean the paradoxes are reinstated. The answer seems to be yes and no” - )

 

3) It has been repeatedly pointed out this Axiom obliterates the distinction according to levels and compromises the vicious-circle principle in the very specific form stated by Russell. But The philosopher and logician FrankRamsey (1903-1930) was the first to notice that the axiom of reducibility in effect collapses the hierarchy of levels, so that the hierarchy is entirely superfluous in presence of the axiom.

 

 

4) Russell abandoned this axiom and many believe it is illegitimate and must

be not used in mathematics

 

Ramsey says

 

Such an axiom has no place in mathematics, and anything which cannot be

proved without using it cannot be regarded as proved at all.

 

This axiom there is no reason to suppose true; and if it were true, this

would be a happy accident and not a logical necessity, for it is not a

tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

[uncool]

Essentially Godel's theorem says you can't have unlimited scope in having statements talk about themselves. To put it a bit more rigorously,

 

Assume a system in which a statement is always allowed to refer to its own truth value or provability value. Then, if the system is consistent, there is a statement that cannot be proven.

 

Where do you disagree?

=Uncool-

[das]

Essentially Godel's theorem says you can't have unlimited scope in having statements talk about themselves. To put it a bit more rigorously,

 

Assume a system in which a statement is always allowed to refer to its own truth value or provability value. Then, if the system is consistent, there is a statement that cannot be proven.

 

Where do you disagree?

 

Godel has arrived at this by using statements philosophy says are invalid thus philosophically his claims are philosophically invalid

 

by using statements that refer to themselves in his proof his proof is philosophically invalid

 

Poincare and Russell said such statements just lead to paradox or contradiction

 

 

Poincare outlawed impredicative definitions But the problem of outlawing impredicative definitions was that a lot of useful mathematics would have to be abandoned “ruling out impredicative definitions would eliminate the contradiction from mathematics, but the cost was too great

[uncool]

Godel has arrived at this by using statements philosophy says are invalid thus philosophically his claims are philosophically invalid

 

by using statements that refer to themselves in his proof his proof is philosophically invalid

 

Poincare and Russell said such statements just lead to paradox or contradiction

 

 

Poincare outlawed impredicative definitions But the problem of outlawing impredicative definitions was that a lot of useful mathematics would have to be abandoned “ruling out impredicative definitions would eliminate the contradiction from mathematics, but the cost was too great

 

Godel was simply proving what you say Poincare and Russel said. He is referring to them in order to say you can't have all of them - that is the entirety of his paradox. You have actually accepted his paradox by accepting the fact that you cannot use self-referential statements.

=Uncool-

[das]

Godel was simply proving what you say Poincare and Russel said. He is referring to them in order to say you can't have all of them - that is the entirety of his paradox. You have actually accepted his paradox by accepting the fact that you cannot use self-referential statements.

 

sorry any proof which uses invalid statements is invalid

if you use a contradictory statement to prove statements are contradictory your proof is invalid - even if statements are contradictory

[Mr Skeptic]

Not familiar with proof by negation, are we?

[foodchain]

Essentially Godel's theorem says you can't have unlimited scope in having statements talk about themselves. To put it a bit more rigorously,

 

Assume a system in which a statement is always allowed to refer to its own truth value or provability value. Then, if the system is consistent, there is a statement that cannot be proven.

 

Where do you disagree?

=Uncool-

 

That is a very interesting question. Do you have any thoughts about such in relation to boundary conditions or chaos theory?

[das]

Not familiar with proof by negation, are we?

sorry Godel uses impredicative statements and philosophy says they are invalid and thus his proof is invalid

even godel says they make maths false

 

 

 

"consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false

[uncool]

das...find an explicit statement and proof of Godel's theorem, and then find where it is wrong. I dare you. Find the exact spot where it is wrong.

=Uncool-

[das]

das...find an explicit statement and proof of Godel's theorem, and then find where it is wrong. I dare you. Find the exact spot where it is wrong.

 

i dont have to all have to do is take godels word for it that he makes statements that refer to themselves

“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves,…

 

you try finding these statements he says they are there -and they make his proof invalid philosophically

[Reaper]

i dont have to all have to do is take godels word for it that he makes statements that refer to themselves

 

 

you try finding these statements he says they are there -and they make his proof invalid philosophically

 

Just....wow. You go from mere speculation to outright denial. "I don't have to know what it is, its wrong because I don't understand it". Classic

[das]

Just....wow. You go from mere speculation to outright denial. "I don't have to know what it is, its wrong because I don't understand it". Classic

sorry i know why he is invalid

he tells us he uses impredicative statements and these are philosophically invalid -thus making his proof invalid

[Reaper]

sorry i know why he is invalid

he tells us he uses impredicative statements and these are philosophically invalid -thus making his proof invalid

 

Then you might want to read this, because impredicativity is absolutely necessary in mathematics in general. Just because it is self referencing does not mean that it is invalid.

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

[das]

Then you might want to read this, because impredicativity is absolutely necessary in mathematics in general. Just because it is self referencing does not mean that it is invalid.

 

go read any book on philosophy and they all say circular statements ie petitio

principi are invalid

 

http://philosophy.lander.edu/logic/circular.html

 

. Petitio Principii: (circular reasoning, circular argument, begging the question) in general, the fallacy of assuming as a premiss a statement which has the same meaning as the conclusion.

 

 

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

In logic, begging the question describes a type of logical fallacy, petitio principii, in which the conclusion of an argument is implicitly or explicitly assumed in one of the premises.[1] Stephen Barker explains the fallacy in The Elements of Logic: "If the premises are related to the conclusion in such an intimate way that the speaker and listeners could not have less reason to doubt the premise than they have to doubt the conclusion, then the argument is worthless as a proof, even though the link between premises and conclusion may have the most cast-iron rigor".[1] In other words, the argument fails to prove anything because it applies what it is supposed to prove as fact

 

even godel said they made mathematics false

 

"consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false.

any first year phil students know this look on the net and you will see

 

Poincare outlawed impredicative definitions But the problem of outlawing impredicative definitions vas that a lot of useful mathematics would have to be abandoned “ruling out impredicative definitions would eliminate the contradiction from mathematics, but the cost was too great " (B, Bunch, op.cit p.134)

[uncool]

das - you do realize that a specific find of where the line is wrong will boost your case so much that seriously, you would get a Fields medal for finding such a line. A proof that Godel's theorem is wrong would get you huge acclaim in the math world.

=Uncool-

[Mr Skeptic]

To das:

Impredicative definitions are not the same as circular reasoning. Circular reasoning is basically an obfuscated "Assume A. Blah, blah, blah. Therefore A." Impredicative definitions are like recursive functions (which I assure you, work perfectly well).

[Reaper]

go read any book on philosophy and they all say circular statements ie petitio

principi are invalid

 

http://philosophy.lander.edu/logic/circular.html

 

 

But the incompleteness theorems aren't using circular logic. Far from it actually. Look at your other thread about the incompleteness theorems, I posted a link about it there. I'll post a wiki link to it here too: http://en.wikipedia.org/wiki/Incompleteness_theorems

 

 

 

even godel said they made mathematics false

 

No he did not. He said that mathematics is incomplete, hence the name "Godel's Incompleteness Theorems". But just because something is incomplete doesn't mean it is false.

 

any first year phil students know this look on the net and you will see

 

Oh really? Since you claim to know it, you then presumably know that one of the basic no-no's of philosophical thought is unsubstantiated claims and/or baseless assumptions. Many philosophy students know this, look on the net and you will see, or better yet spend more time studying philosophy and the theory you are trying to disprove and then we will talk.

[das]

No he did not. He said that mathematics is incomplete, hence the name "Godel's Incompleteness Theorems". But just because something is incomplete doesn't mean it is false.

 

sorry godel states

 

"consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false”

 

But the incompleteness theorems aren't using circular logic

godel is useing circular statement and philosophy poncicare russell et al say they are invalid

Quote from Godel

“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves,…

 

What Godel understood by "propositions which make statements about

themselves"

 

is the sense Russell defined them to be

 

'Whatever involves all of a collection must not be one of the collection.'

Put otherwise, if to define a collection of objects one must use the total

collection itself, then the definition is meaningless. This explanation

given by Russell in 1905 was accepted by Poincare' in 1906, who coined the

term impredicative definition, (Kline's "Mathematics: The Loss of

Certainty"

 

Impredicative definitions are like recursive functions (which I assure you, work perfectly well

 

sorry impredicative statements are as godel ponicare and russel said they are said they are

 

We saw that we can construct propositions which make statements about themselves

 

and they are invalid as godel says they are

"consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false”

[bascule]

Can an admin please lock this thread? This person is a known multi-forum troll spreading complete bullshit.

 

Here's some of my canned responses to this asshole:

 

Godel's second theorem applies to any formal system extending Peano arithmetic (the P axioms)

 

The simple theory of types that emerges from Russel's Axiom of Reducibility does not undermine Godel's second theorem. Russel instead underestimated the expressive power of type systems emergent from the natural numbers:

 

http://www.ams.org/notices/200604/fea-kanamori.pdf

 

Godel's advances in set theory can be seen as part of a steady intellectual development from his fundamental work on incompleteness. His 1931 paper had a prescient footnote 48a:

 

As will be shown in Part II of this paper, the true reason for the incompleteness inherent in all formal systems of mathematics is that the formation of ever higher types can be continued into the transfinite (cf. D. Hilbert, Uber das Unendliche, Math. Ann. 95, p. 184), while in any formal system at most countably many of them are available. For it can be shown that the undecidable propositions constructed here become decidable whenever appropriate higher types are added. An analogous situation prevails for the axiom system of set theory.

 

 

http://www.springerlink.com/content/kw175702q720007t/

 

...the true reason for the incompleteness inherent in all formal systems of mathematics is that the formation of ever higher types can be continued into the transfinite, while in any formal system at most countably many of them are available.

 

Russel's type system and its surrounding axioms are irrelevant since infinite types are emergent from his logic system and furthermore infinite types are required for a complete system, yet the axiom of reducibility limits PM to finite types. To the contrary of your (vicarious) argument, the axiom of reducibility necessitates an incomplete system. Godel's non-usage of it was irrelevant, since the requisite type system was emergent from the Peano axioms alone (per the second incompleteness theorem).

 

Including the Axiom of Reducibility in PM necessitates it be relegated to an incomplete system since Godel proved a complete system requires infinite types. Omitting AR does not make for a complete system either since the provision of an infinite type system (per the second completeness theorem) facilitates self-referential statements which are unprovable in the system in which they are declared.

 

Can you please state a formal refutation of Godel's proof in predicate calculus?

 

Anything else is mere sophistry and evidence that you don't know what the f*ck you're talking about.

[Phi for All]

I suspect das has earwax on his fingertips. I can almost hear him singing....

 

 

 

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