simple paradox in Godels incompleteness theorem that invalides it

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pam69ur
 
Reply Thu 4 Oct, 2007 11:35 pm
The Australian philosopher colin leslie dean points out a paradox in Godels incompleteness theorem that invalidate it ie makes it a complete failure and meaningless

extracted from his book-posted below

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[/CENTER]
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[CENTER][CENTER]CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS[/CENTER]
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[CENTER][CENTER]By[/CENTER]
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[CENTER][CENTER]COLIN LESLIE DEAN[/CENTER]
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[CENTER][CENTER]B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,[/CENTER]
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[CENTER][CENTER] M.A (PSYCHOANALYTIC STUDIES), MASTER OF PSYCHOANALYTIC STUDIES, GRAD CERT (LITERARY STUDIES)[/CENTER]
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[CENTER][CENTER]CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS[/CENTER]
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[CENTER][CENTER]By[/CENTER]
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[CENTER][CENTER]COLIN LESLIE DEAN[/CENTER]
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[CENTER][CENTER]B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,[/CENTER]
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[CENTER][CENTER] M.A (PSYCHOANALYTIC STUDIES), MASTER OF PSYCHOANALYTIC STUDIES, GRAD CERT (LITERARY STUDIES)Read criticism section first starting at page 17-20 part 2, then back to 14 part 1 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 The undecidable However if a formula and its own negation are both formally demonstrable the mathematical calculus is not consistent Rather than accept a self-contradiction mathematicians settle for the second choiceTO GIVE DETAILThe 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
"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" - Mind -- Sign In Page
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 )

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)


AXIOM OF CHOICE
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 (ACBanach-Tarski Paradox "- http://www.math.vanderbilt.edu/~schectex/ccc/choice.html)

ZERMELO AXIOM SYSTEM

Godel specifies that he uses the Zermelo axiom system- (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.28.)

quote
Godel's first Incompleteness Proof at MROB at MROB


"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

[191]a system made by adding a recursively definable ω-consistent class of axioms. As can be easily confirmed, the systems which satisfy assumptions 1 and 2 include the Zermelo-Fraenkel and the v. Neumann axiom systems of set theory,47"

IMPREDICATIVE DEFINITIONS
Godel used impredicative definitions

Quote from Godel
(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 andmake it false"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 secondincompleteness theorems
(K Godel , On undecidable propositions of formal mathematical systems in The undecidable Preintuitionism - Wikipedia, the free encyclopediaGODEL ACCEPTED IMPREDICATIVE DEFINITIONS
quote
"consider this rather as a proof that the vicious circle principle is false than that classical mathematics is falseTHEORY OF TYPES
In Godels second incompleteness theorem he uses the theory of types- but with out the very axiom of reducibility that was required to avoid the serious problems with the theory of types and to make the theory of types work.- without the axiom of reducibility virtually all mathematics breaks down. (PlanetMath: Russell's theory of typesWe shall depend on the theory of types as our means for avoiding paradox. .Accordingly we exclude the use of variables running over all objects and use different kinds of variables for different domians. Speciically p q r... shall be variables for propositions . Then there shall be variables of successive types as followsx y z for natural numbersf g h for functionsDifferent formal systems are determined according to how many of thesetypes of variable are used... (K Godel , On undecidable propositions of formal mathematical systems in The undecidable , M, Davis, Raven Press, 1965, p.63 of this work Davis notes, "it covers ground quite similar to that covered in Godels orgiinal 1931 paper on undecidability," p. 46.). Clearly Godel is using the theory of types as part of his meta-theory to show something in his object theory i.e. his formal system example.

Russell propsed the system of types to eliminate the paradoxes from mathematics. But the theory of types has many problems and complications .One of the devices Russell used to avoid the paradoxes in his theory of types was to produce a hierarchy of levels. A big problems with this device , is that the natural numbers have to be defined for each level and that creates insuperable difficulties for proofs by inductions on the natural numbers where it would more convenient to be able to refer to all natural numbers and not only to all natural numbers of a certain level. This device makes virtually all mathematics break down. (PlanetMath: Russell's theory of typesBut in the second incompleteness theorem Godel does not use the very axiom of reducibility Russell had to introduce to avoid the serious problems with the theory of types. Thus he uses a theory of types which results in the virtual breakdown of all mathematics
(http://www.helsinki.fi/filosofia/gts/ramsay.pdf) (PlanetMath: Russell's theory of types)






[CENTER][CENTER]GODEL IS SELF-CONTRADICTORY[/CENTER]
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But here is a contradiction Godel must prove that asystem 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
[CENTER][CENTER]CRITICISMS[/CENTER]
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Some say Godel did not use the axioms of choice and the axiom of reducibility in he incompleteness theorems

Others say he only used the axiom of reducibility in his object theory but not his meta-theory

Godels statements indicate that he did use AR and AC in both his meta-theory and so called object theory

If he did not use all axioms of the systems of PM then when he states

"we now show that the proposition [R(q);q] is undecidable in PM" (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable The undecidable , M, Davis, Raven Press, 1965, p.6)





Godel uses the axiom of reducibility and axiom of choice from the PM
he states
"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)

on page 7 he states ((K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965)
"now we obtain an undecidable proposition of the system PM"

Clearly this undecidable proposition comes about due the axioms etc which PM uses

Godel goes on
"the ternary relation z=[y;z] also turns out to be definable in PM" (ibid, p,8)

Godel goes on
"since the concepts occurring in the definiens are all definable in PM" (ibid,p.8)

Godel has told us PM is made up of axiom of reducibility, axiom of choice etc so
these definiens must be defined interms of these axioms

Godel goes on
"we now show that the proposition [R(q);q] is undecidable in PM"(K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.8)) - again this must mean undecidable within PMs system ie its axioms etc

further
Godel e goes on
"we pass now to the rigorous execution of the proof sketched above and we first give a precise description of the formal system P for which we wish to prove the existence of undecidable propositions" (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.9)

Some call this system P the object theory but they are wrong in part
for Godel goes on
"P is essentially the system which one obtains by building the logic of PM around Peanos axioms..." K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,, p.10)

Thus P uses as its meta-theory the system PM ie its axioms of choice reducibility etc (he has told us this is what PM SYSTEM IS)

Thus P is made up of the meta-theory of PM and Peanos axioms

Thus by being built on the meta-theory of PM it must use the axioms of PM
etc and these axioms are choice reducibility etc

If godel tells us he is going to using the axioms of PM but only use some
of them in fact then he is both wrong and lying when he tells us that
"we now show that the proposition [R(q);q] is undecidable in PM" K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,,p. 8)

and
"the proposition undecidable in the system PM is thus decided by
metamathemaical arguments" K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,, p.9)




Thus simply
Godel tells us
1) he is using the axioms of PM
2) the proposition is undecidable in the system PM
2)P uses as its meta-system the axioms of PM
3) so the proof in P must use PMs axioms
3) if he does not use all the axioms of PM then he is lying to us when he
say "there are undeciable propositions in PM, and P

So is Godel lying on these points
As I have argued the axioms he uses are invalid and flawed thus making his theorems invalid flawed and a complete failure

[CENTER][CENTER]2[/CENTER]
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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 propositionsThe undecidable , M, Davis, Raven Press, 1965, p.63) But the theory of types cannot over come the syntactical paradoxes i.e. liar paradox." (E, Carniccio op.cit, p.345.) Also this procedure created unending problems such that Russell had to introduce his axiom of reducibility ( Bunch, op.cit, p,.135). But even though the axiom with the theory of types created results that don't fall into any of the known paradoxes it leaves doubt that other paradoxes want crop up. But this axiom is so artificial and create a whole nest of other problems for mathematics that Russell eventually' abandoned it (Bunch, ibid, p.135.) Godel uses this axiom in his impossibility' proof. (K. Godel, op.cit, p.5) "Thus these attempts to solve the paradoxes all turned out to involve either paradoxical notions them selves or to artificial that most mathematicians rejected them
[CENTER][CENTER]AXIOM OF CHOICESKOLEM PARADOX[/CENTER]
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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"
from
http://www.earlham.edu/~peters/courses/logsys/low-skol.htm
Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?



from

http://en.wikipedia.org/wiki/Skolem's_paradoxhttp://www.earlham.edu/~peters/courses/logsys/low-skol.htm
Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?

Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it

attempted solutions
Bunch notes


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



http://www.earlham.edu/~peters/courses/logsys/low-skol.htm

One reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction

The good news is that this strongly paradoxical reading is optional. The bad news is that the obvious alternatives are very ugly. The most common way to avoid the strongly paradoxical reading is to insist that the real numbers have some elusive, essential property not captured by system S. This view is usually associated with a Platonism that permits its proponents to say that the real numbers have certain properties independently of what we are able to say or prove about them.

The problem with this view is that LST proves that if some new and improved S' had a model, then it too would have a countable model. Hence, no matter what improvements we introduce, either S' has no model or it does not escape the air of paradox created by LST. (S' would at least have its own typographical expression as a model, which is countable.

then the faith solution


Finally, there is the working faith of the working mathematician whose specialization is far from model theory. For most mathematicians, whether they are Platonists or not, the real numbers are unquestionably uncountable and the limitations on formal systems, if any, don't matter very much. When this view is made precise, it probably reduces to the second view above that LST proves an unexpected limitation on formalization. But the point is that for many working mathematicians it need not, and is not, made precise. The Skolem paradox has no sting because it affects a "different branch" of mathematics, even for mathematicians whose daily rounds take them deeply into the real number continuum, or through files and files of bytes, whose intended interpretation is confidently supposed to be univocal at best, and at worst isomorphic with all its fellow interpretations.




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perplexity
 
Reply Fri 5 Oct, 2007 02:25 am
@pam69ur,
I'd thought we already knew that for many working mathematicians it need not and is not made precise.

Why then the 7 thousand words of cut and paste to several different forums?

Does the original author permit this annoyance, or is it a flagrant breach of copyright?

:confused:

The problem simply put is that science itself pretends to be an established philosophical truth but is in fact profoundly religious.

:p
 
 

 
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