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5 Reasons why Godels incompleteness theorem invalid
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Posted by: damsell
The Autralian philosopher Colin Leslie Dean points out
Godels theorems are invalid for 5 reasons: he uses the axiom of
reducibility- which is invalid, he uses the axiom of choice, he
constructs impredicative statements - which are invalid ,he miss uses
the theory of types, he falls into 3 paradoxes
http://gamahucherpress.yellowgum.co...ophy/GODEL5.pdf
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. Godels theorems are invalid for 5 reasons: he uses the axiom of reducibility- which is invalid, he uses the axiom of choice, he constructs impredicative statements - which are invalid ,he miss uses the theory of types, he falls into 3 paradoxes (Read criticism section first starting at page 19-28 part 2, then back to 16 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 – and impredicative definitions thus showing that Godel’s proof is based upon a misguided system of axioms and impredicative definitions and that it is invalid as its axioms and impredicative definitions 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 and impredicative definitions
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. All that Gödel proved was the lair paradox
Gödel used impedicative definitions- Russell and Ponicare rejected these as they lead to paradox
Godel , K , On undecidable propositions of formal mathematical systems, in The undecidable , M, Davis, Raven Press, 1965, p.63 )
Gödel used the axiom of reducibility -Russell abandoned this –some say it leads 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 and impredicative definitions only succeeded in getting the inevitable paradox that his axioms and impredicative definitions ordained him to get. In other words he could have only ended in paradox for this is what his axioms and impredicative definitions 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 and impredicative definitions logically would lead him to; because these axioms and impredicative definitions lead to or end in paradox themselves. All he succeeded in getting was a paradoxical result.. Godel by using those axioms and impredicative definitions he could only have 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
“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
“ [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u & u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]
x is a formula derived from the axiom-schema IV, 1 by substitution “ http://www.mrob.com/pub/math/goedel.html
( 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/...107/428/823.pdf
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)
4) Russell Ramsey and Witgenstein regarded it as illegitimate 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
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/~sch...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
http://www.mrob.com/pub/math/goedel.html
"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
Ponicare Russell and philosophers argue these types of definitions are invalid Ponicare Russell point out that they lead to contradictions in mathematics
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
http://www.friesian.com/goedel/chap-1.htm
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 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 used Peanos axioms but these axioms are impredicative and thus according to Russell Poincaré and others must be avoided as they lead to paradox.
quote
http://en.wikipedia.org/wiki/Preintuitionism
”This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano's axiomatic theory of natural numbers.
Peano's fifth axiom states:
* Allow that; zero has a property P;
* And; if every natural number less than a number x has the property P then x also has the property P.
* Therefore; every natural number has the property P.
This is the principle of complete induction, it establishes the property of induction as necessary to the system. Since Peano's axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x+1. What one can do is say that, if after some number n of trails that show a property P conserved in x and x+1, then we may infer that it will still hold to be true after n+1 trails. But this is itself induction. And hence the argument is a vicious circle.
From this Poincaré argues that if we fail to establish the consistency of Peano's axioms for natural numbers without falling into circularity, then the principle of complete induction is improvable by general logic. “
GODEL ACCEPTED IMPREDICATIVE DEFINITIONS
quote
http://www.friesian.com/goedel/chap-1.htm
”recent research [9] has shown that more can be squeezed out of these restrictions than had been expected:
all mathematically interesting statements about the natural numbers, as well as many analytic statements, which have been obtained by impredicative methods can already be obtained by predicative ones.[10]
We do not wish to quibble over the meaning of "mathematically interesting." However, "it is shown that the arithmetical statement expressing the consistency of predicative analysis is provable by impredicative means." Thus it can be proved conclusively that restricting mathematics to predicative methods does in fact eliminate a substantial portion of classical mathematics.[11]
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."[12]”
THEORY 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. (http://planetmath.org/encyclopedia/...ducibility.html)
As he states “ We now describe in some detail a formal system which will serve as an example for what follows …We 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 follows
x y z for natural numbers
f g h for functions
Different formal systems are determined according to how many of these
types 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. (http://planetmath.org/encyclopedia/...ducibility.html) For example, when speaking of real numbers system and its completeness, one wishes to quantify over all predicates of real numbers…, not only of those of a given level. In order to overcome this, Russell and Whitehead introduced in PM the so-called axiom of reducibility – but as we have seen 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 Frank Ramsey (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. But 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) (http://planetmath.org/encyclopedia/...ducibility.html)
Godels theorems are invalid for 5 reasons: he uses the axiom of reducibility- which is invalid, he uses the axiom of choice, he constructs impredicative statements - which are invalid ,he miss uses the theory of types, he falls into 3 paradoxes Gödel is a complete failure as he ends in utter meaninglessness. His meaningless/paradoxical result comes directly from using axioms and impredicative definitions that lead or end in paradox. Even if Godel did not prove that mathematics was inconsistent Gödel proved nothing as it was totality built upon invalid axioms and impredicative definitions; All talk of what Godel achieved is just another myth mathematicians foist upon an ignorant population to beguile them into believing mathematician know what they are talking about and have access to truth.
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
CRITICISMS
1
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 , M, Davis, Raven Press, 1965, p.8)
he must have been lying
Godels states
quote
“ 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, 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
2
There are 3 paradoxes in Godels proof
1 paradox
Godel makes the claim that there are undecidable propositions in a constructed system [PM and ZF] that dont depend upon the special nature of the constructed system [PM and ZF]
Quote
As he states
“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 that this is not the case but rather that in both systems cited [PM and ZF] there exist relatively simple problems of ordinary whole numbers [undecidability] which cannot be decided on the basis of the axioms. [NOTE IT IS CLEAR] This situation [ undecidability which cannot be decided on the basis of the axioms]. does not depend upon the special nature of the constructed systems [PM and ZF] but rather holds for a very wide class of formal systems among which are included in particular all those which arise from the given systems [PM and ZF] by addition of finitely many axioms” (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)
Thus Godel says he is going to show that undecidability is not dependent on the
axioms of a system or the speacial nature of PM and ZF
Also
Godels refers to PM and ZF AS FORMAL SYSTEMS
"the most extensive formal systems constructed .. are PM ZF" ibid, p.5
so when he states that
"This situation does not depend upon the special nature of the constructed
systems but rather holds for a very wide class of formal systems"
he must be refering to PM and ZF as belonging to these class of formal systems- further down you will see this is true as well
thus he is saying
the undecidability claim is independent of the axioms of the formal system but PM is a formal system
Godel says he is going to show undecidabilitys 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 V1 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 [ which uses system PM] and is w - consistent there exist undecidable propositions ”. (ibid, p.28)
CLEARLY GODEL IS MAKING SWEEPING CLAIMS JUST BASED UPON HIS P PROOF
Godel has said that undecidability is not dependent on the
axioms of a system or the special nature of PM and ZF
There is a paradox here
He says every formal system which satisfies assumption 1 and 2 ie
based upon axioms - but he has said undecidablity is independent of axioms
2 paradox
Also there is a contradiction here
Godel has said undecidablity is not dependent on PM yet says it is hence” in every formal system which satisfies assumptions 1 and 2 [ which uses system PM] and is w - consistent there exist undecidable propositions “
Thus the paradox undedciablity is not dependent of the axioms of a system or PM but is
dependent on the axioms of the system and PM
In the above Godel must be referring to PM and ZF as they are formal systems
but he has said
"This situation does not depend upon the special nature of the constructed
systems [PM ZF] but rather holds for a very wide class of formal systems"
now P is constructed with the axioms of PM and Peano axioms
"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)
so clearly undecidability is dependent on the quirky nature of PM-which is a formal system
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
QUOTE
[undecidability]does not depend upon the special nature of the
constructed systems [PM and ZF] but rather holds for a very wide class of formal systems
contradict this
hence in every formal system which satisfies assumptions 1 and 2 [depending on the special nature of formal system P WHICH USES PM ] and is w - consistent there exist undecidable propositions
HE HAS SAID UNDECIDABILITY DOES NOT DEPEND UPON THE NATURE OF PM YET SAYS UNDECIABILITY IN FORMAL SYSTEMS- OF WHICH PM- IS ONE IS DEPENDENT ON PM
put simply
Undecidability is independent on nature of PM, yet is dependent on the nature of PM.
thus undecidability is not dependent on the nature of the [PM and ZF] but he has said undecidability is dependent upon the nature of formal system P which uses PM
thus
“[undecidability] does not depend upon the special nature of the
constructed systems [PM and ZF] but rather holds for a very wide class of formal systems “
Contradicts this
“hence in every formal system which satisfies assumptions 1 and 2 [ depends upon the special nature of formal system PM] and is w - consistent there exist undecidable propositions ”.
Thus when Godel states
"hence in every formal system [PM example] which satisfies assumptions 1
and 2 and is w [Dependent on the special nature of P and thus PM ] -
consistent there exist undecidable propositions"
he is creating paradox and circularity of argument
he says undecidability is independent of formal system PM and ZF yet
deriving assumptions dependent on this formal system PM he says those
formal systems that have these assumption have undecidability and he
states ZF has these assumptions (ibid, p.28)
put simply
Undecidability is independent on nature of PM, yet is dependent on the nature of PM.
clearly Godel is in paradox and invalid due to meaninglessness
3 paradox
There is another paradox in Godels incompleteness theorem
As we have seen undecidability in a formal system is dependent on the system PM but the system PM has undecidability
Godel tells us that among those very wide range of formal systems that have undecidability are to be included those systems which arise from PM by the addition finitely many axioms
As he states
“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 that this is not the case but rather that in both systems cited [PM and ZF] 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 [PM and ZF] but rather holds for a very wide class of formal systems among which are included in particular all those which arise from the given systems [PM and ZF] by addition of finitely many axioms”
In other words PM is included in those systems which have undecidablity
Thus we have the paradox that while PM is used to find if a formal system is undecidable it is undecidable itself
i.e.
hence in every formal system which satisfies assumptions 1 and 2 [ from P which uses system PM] and is w - consistent there exist undecidable propositions
In other words the very system which is used to find undecidability is included in the set of undecidable systems
PM is part of the very set it is used to create
Gödel's proof shows for some class of formal systems, they can not be both complete and consistent
if a system is consistent it will be incomplete
If PM is consistent it is incomplete i.e it has statements which cannot be proven true or false
thus
PM is used to prove that a system has statements which cannot be proven true or false
but
PM can only prove this if all its statement can be proven to be true
but
PM has statements which cannot be proven true or false
thus
it cant prove anything
but it is used to prove if systems are undecidadble
thus a paradox
PM being undecidable cant be used to create the set of undecidable systems of which it belongs-if it belongs to the set it cant prove anything and if it dont belong to the set it is not undecidable[/b]
Thus we have the situation overall that clearly Godel is in paradox and invalid due to meaninglessness
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- which includes PM which is itself undecidable
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. Godels theorems are invalid for 5 reasons: he uses the axiom of reducibility- which is invalid, he uses the axiom of choice, he constructs impredicative statements - which are invalid ,he miss uses the theory of types, he falls into 3 paradoxes
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