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Author Topic: A paradox in Godels incompleteness theorem that invalidates it  (Read 3663 times)

Offline bettybop

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The australian philosopher colin leslie dean points out a simple paradox in Godels incompleteness theorem that invalidate it and makes it a complete failure

extracted from his book
Godel makes the claim that there are undecidable propositions in a formal system that dont depend upon the special nature of the formal system
Quote

It is reasonable therefore to make the conjecture that these axioms and rules of inference are also sufficent to decide all mathematical questions which can be formally expressed in the given systems. In what follows it will be shown .. there exist relatively simple problems of ordinary whole numbers which cannot be decided on the basis of the axioms. [NOTE IT IS CLEAR] This situation does not depend upon the special nature of the
constructed systems but rather holds for a very wide class of formal systems (K Godel , On formally undecidable propositions of principia mathematica and related
systems in The undecidable , M, Davis, Raven Press, 1965, p.6).( K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.6)

Godel says he is going to show this by using the system of PM (ibid)
he then sets out to show that there are undecidable propositions in PM (ibid. p.8)

where Godel states
"the precise analysis of this remarkable circumstance leads to surprising results concerning consistence proofs of formal systems which will be treated in more detail in section 4 (theorem X1) ibid p. 9 note this theorem comes out of his system P
he then sets out to show that there are undecidable propositions in his system P -which uses the axioms of PM and Peano axioms.
at the end of this proof he states
"we have limited ourselves in this paper essentially to the system P and have only indicated the applications to other systems" (ibid p. 38)

now
it is based upon his proof of undecidable propositions in P that he draws out broader conclusions for a very wide class of formal systems
After outlining theorem V1 in his P proof - where he uses the axiom of choice- he states
"in the proof of theorem 1V no properties of the system P were used other than the following
1) the class of axioms and the riles of inference- note these axioms include reducibility
2) every recursive relation is definable with in the system of P
hence in every formal system which satisfies assumptions 1 and 2 and is w - consistent there exist undecidable propositions ?. (ibid, p.28)

CLEARLY GODEL IS MAKING SWEEPING CLAIMS JUST BASED UPON HIS P PROOF

but
he has told us undecidable propositions in a formal system are not due to the nature of the formal system but he is making claims about a very wide range of formal systems based upon the nature of formal system P

1) there is circularity/paradox of argument he says his consistency proof is independent of the nature of a formal system yet he bases this claim upon the very nature of a particular formal system P
2) he is clearly basing his claims for his consistency theorems upon the systems PM and P

P and PM are the meta-theories/systems he uses to prove his claim that there are undecidable propositions in a very wide range of formal systems


We have a dilemma
1)either Gödel is right that his claims for undecidability of formal systems
are independent of the nature of a formal system

and thus he is in paradox when he makes claims about formal systems based
upon the special nature of P - AND THUS PM

OR
2) he makes claims about formal systems based upon the special nature of P
and PM
that would mean that PM and P are the meta-systems/meta-theory through
which he is make undecidable claims about formal systems

thus indicating the axioms of PM and P are central to these meta claims
there by when I argue s these axioms are invalid then Godels
incompleteness theorem is invalid and a complete failure.

Thus either way Godels incompleteness theorem are invalid and a complete failure :either due to the paradox in his theorem or the invalidity of his axioms.


to see the arguments that demonstrate the axioms godel uses are invalid soo the following work


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)
 
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

A case  study in the view that   all views end in meaninglessness. As an example of this is Gödel’s incompleteness theorem.  Gödel is a complete failure as he ends in utter meaninglessness. (Read criticism section first starting at page 17-20 part 2, then back to 14 part 1)

What Gödel proved was not the incompleteness theorem but that mathematics was self contradictory. But he proved this with flawed and invalid axioms- axioms that either lead to paradox or ended in paradox –thus showing that Godel’s proof is based upon a misguided system of axioms and that it is invalid as its axioms are invalid. For example Godels uses the axiom of reducibility but this axiom was rejected as being invalid by
Russell as well as most philosophers and mathematicians. Thus just on this point Godel is invalid as by using an axiom most people says is invalid he creates an invalid proof due to it being based upon invalid axioms

Godel states “the most extensive formal systems constructed up to the present time are the systems of Principia Mathematica (PM) on the one hand and on the other hand the Zermel-Fraenkel axiom system of set theory … it is reasonable therefore to make the conjecture that these axioms and rules of inference are also sufficient to decide all mathematical questions which can  be formally expressed in the given axioms. In what follows it will be shown that this is not the case but rather that in both  of the cited systems there exist relatively simple problems of the theory of ordinary numbers which cannot be decided on the basis of the axioms”   (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,pp.5-6)

All that he proved was in terms of PM and Zermelo axioms-there are other axiom systems -so his proof has no bearing outside that system he used Russell rejected some axioms he used as they led to paradox. All that Gödel proved was the lair paradox -which Russell said would happen

Gödel used impedicative definitions- Russell rejected these as they lead to paradox  (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.63)
Gödel used the axiom of reducibility -Russell abandoned this as it lead to paradox (K. Godel, op.cit, p.5)   


Gödel used the axiom of choice mathematicians still hotly debate its validity- this axiom leads to the Branch-Tarski and Hausdorff paradoxes (K.Godel, op.cit, p.5)
Gödel used Zermelo axiom system but this system has the skolem paradox which reduces it to meaninglessness or self contradiction
Godel proved that mathematics was inconsistent
from Nagel -"Gödel" Routeldeg & Kegan, 1978, p 85-86


Gödel also showed that G is demonstrable if and only if it’s formal negation ~G is demonstrable. However if a formula and its own negation are both formally demonstrable the mathematical calculus is not consistent (this is where he adopts the watered down version noted by bunch) accordingly if (just assumed to make math’s consistent) the calculus is consistent neither G nor ~G is formally derivable from the axioms of mathematics. Therefore if mathematics is consistent G is a formally undecidable formula Gödel then proved that though G is not formally demonstrable it nevertheless is a true mathematical formula


From Bunch
"Mathematical fallacies and paradoxes” Dover 1982"  p .151

Gödel proved

~P(x,y) & Q)g,y)
in other words ~P(x,y) & Q)g,y) is a mathematical version of the liar paradox. It is a statement X that says X is not provable. Therefore if X is provable it is not provable a contradiction. If on the other hand X is not provable then its situation is more complicated. If X says it is not provable and it really is not provable then X is true but not provable Rather than accept a self-contradiction mathematicians settle for the second choice


Thus Godel  by using invalid axioms i.e. those that lead to paradox or end in paradox only succeeded in getting the inevitable paradox that his axioms ordained him to get. In other words he could have only ended in paradox   for this is what his axioms determined him to get. Thus his proof   is a complete failure as his   proof. that   mathematics is inconsistent   was the only result that he could have logically arrived at since this result is what his axioms logically would lead him to; because these axioms lead to or end in paradox themselves.  All he succeeded in  getting was  a paradoxical result as Russell new would happen if those axioms where used.  Godel   by using those axioms could only   arrived at a paradoxical result


Gödel used the  Zermelo axiomatic system but this system end in  meaninglessness. There  is the Skolem paradox which collapses axiomatic theory into meaningless

Bunch notes  op cit p.167

“no one has any idea of how to re-construct axiomatic set theory so that this paradox does not occur”



TO GIVE DETAIL
Godel states that he is going to use the system of PM
“ before  we go into details lets us first sketch the main ideas of the proof … the formulas of a formal    system (we limit ourselves here to the system PM) …” ((K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,pp.-6)

Godel uses the axiom of reducibility and axiom of choice from the PM

Quote
http://www.mrob.com/pub/math/goedel.htm [nofollow]
“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition, Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), and the axioms of reducibility and of choice (for all types)” ((K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.5)

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” - http://mind.oxfordjournals.org/cgi/reprint/107/428/823.pdf [nofollow]
 2)“does this mean the paradoxes are reinstated. The answer seems to be yes and no” - http://fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf [nofollow]  )

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.
(http://www.helsinki.fi/filosofia/gts/ramsay.pdf [nofollow])


AXIOM OF CHOICE
Godel states he uses the axiom of choice “this allows us to deduce that even with the aid of the axiom of choice (for all types) … not all sentences are decidable…” (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965. p.28.) Quite clearly the axiom of choice is part of the meta-theory  used in the deduction
(“The Axiom of Choice (AC) was formulated about a century ago, and it was controversial for a few of decades after that; it may be considered the last great controversy of mathematics…. A few pure mathematicians and many applied mathematicians (including, e.g., some mathematical physicists) are uncomfortable with the Axiom of Choice. Although AC simplifies some parts of mathematics, it also yields some results that are unrelated to, or perhaps even contrary to, everyday "ordinary" experience; it implies the existence of some rather bizarre, counterintuitive objects. Perhaps the most bizarre is the Banach-Tarski Paradox “– http://www.math.vanderbilt.edu/~schectex/ccc/choice.html [nofollow])



IMPREDICATIVE DEFINITIONS
Godel used impredicative definitions

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.)

Godels has argued that impredicative definitions destroy mathematics and
make it false

http://www.friesian.com/goedel/chap-1.htm [nofollow]

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”

Yet Godel uses impredicative definitions in his first and second
incompleteness theorems

“ 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.)





GODEL IS SELF-CONTRADICTORY
But here is  a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If   the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent.  But if his proof is true then he has proved that the logic he uses to   make the proof must be consistent, but his proof proves that this cannot be done


























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Offline dkv

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What kind of framework in your opinion dissolves the undecidability or inconsistency ?
Do you have any alternative?
The proof is valid for any arbitary system.
Can you describe a logical system in which the contradiction can not take place and rest of mathematics remains the same?

 

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