G?DEL'S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS G?DEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS
CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS

B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,
M.A (PSYCHOANALYTIC STUDIES), GRAD CERT(LITERARY STUDIES)

G?DEL'S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS G?DEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS
CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS


B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,
M.A (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 G?del's incompleteness theorem. G?del is a complete failure as he ends in utter meaninglessness

What G?del 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

All that he proved was in terms of 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 G?del proved was the lair paradox -which Russell said would happen

G?del 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)
G?del used the axiom of reducibility -Russell abandoned this as it lead to paradox (K. Godel, op.cit, p.5)

G?del 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)

G?del used Zermelo axiom system but this system has the skolem paradox which reduces it to meaninglessness or self contradiction

from Nagel -"G?del" Routeldeg & Kegan, 1978, p 85-86


G?del 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 G?del 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

G?del 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


G?del 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"



G?del is a complete failure as he ends in utter meaninglessness G?del proved nothing as it was totality built upon invalid axioms; it 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

Appendix
IMPREDICATIVE DEFINITIONS
AXIOM OF REDUCIBILITY

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) Also as Russell pointed cut the notion of impredicative definitions was paradoxical as the property applies to itself "is the property . of being impredicative itself impredicative or not" (this is another analog of Gretling's paradox.) (ibid, p.134.). Russell tried to solve the paradoxes by his theory of types Russell and Whitehead explained the logical antinomies as Being due to a vicious circle their theory of types 'was means to irradiate these vicious circles by, making them by definition not allowed ( E, Carnuccio , Mathematics and logic in history and contemporary thought, Faber & Faber 1964, 344-355.)-[ but Godel sayys be disagrees with Russell and uses them in his impossibility, proof] (K Godel , On formally undecidable propositions of principia mathematica and related systems in The 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
AXIOM OF CHOICE

Godel used the axiom of choice in his impossibility proof (K.Godel, op.cit, p.5)" But ever since its use by Zermelo there have been problems with this axiom "Cohen proved that he axiom of choice is independent of the other axioms of set l theory. As a result you can have Zermeloian mathematics that accept the
axiom of choice or various non-Zermeloian mathematics that reject it in one way or another? Cohen also proved that there is a Cantorian mathematics in which the continuum hypothesis is true and a non-Cantorian mathematics in which it is denied (B, Bunch, op.cit, p.169). If the axiom of choice is kept then we get the BranchTarski and Hausdorff paradoxes Now "mathematicians who have thought about it have decided that the Branch-Traski is one of the paradoxes that "you just live with it" (ibid, p.180.) As Bunch notes "rejection of the axiom of choice means rejection of Important parts of "classical." mathematics and set theory. Acceptance of the axiom of choice however has some peculiar implications of its own i e Branch-Tarski and Hausdorff paradoxes (ibid,p. 169-170).


SKOLEM PARADOX

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/cours...s/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

Skolem's paradox - Wikipedia, the free encyclopedia
the "paradox" is viewed by most logicians as something puzzling, but not a paradox in the sense of being a logical contradiction (i.e., a paradox in the same sense as the Banach-Tarski paradox rather than the sense in Russell's paradox). Timothy Bays has argued in detail that there is nothing in the L?wenheim-Skolem theorem, or even "in the vicinity" of the theorem, that is self-contradictory.

However, some philosophers, notably Hilary Putnam and the Oxford philosopher A.W. Moore, have argued that it is in some sense a paradox.

The difficulty lies in the notion of "relativism" that underlies the theorem. Skolem says:

In the axiomatization, "set" does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set M of the domain B is nondenumerable in the sense of the axiomatization; for this means merely that within B there occurs no one-to-one mapping of M onto Z0 (Zermelo's number sequence). Nevertheless there exists the possibility of numbering all objects in B, and therefore also the elements of M, by means of the positive integers; of course, such an enumeration too is a collection of certain pairs, but this collection is not a "set" (that is, it does not occur in the domain B).

Moore (1985) has argued that if such relativism is to be intelligible at all, it has to be understood within a framework that casts it as a straightforward error. This, he argues, is Skolem's Paradox

Zermelo at first declared the Skolem paradox a hoax. In 1937 he wrote a small note entitled "Relativism in Set Theory and the So-Called Theorem of Skolem" in which he gives (what he considered to be) a refutation of "Skolem's paradox", i.e. the fact that Zermelo-Fraenkel set theory --guaranteeing the existence of uncountably many sets-- has a countable model. His response relied, however, on his understanding of the foundations of set theory as essentially second-order (in particular, on interpreting his axiom of separation as guaranteeing not merely the existence of first-order definable subsets, but also arbitrary unions of such). Skolem's result applies only to the first-order interpretation of Zermelo-Fraenkel set theory, but Zermelo considered this first-order interpretation to be flawed and fraught with "finitary prejudice". Other authorities on set theory were more sympathetic to the first-order interpretation, but still found Skolem's result astounding:

* At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known. (John von Neumann)

* Skolem's work implies "no categorical axiomatisation of set theory (hence geometry, arithmetic [and any other theory with a set-theoretic model]...) seems to exist at all". (John von Neumann)

* Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached. (Abraham Fraenkel)

* I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique. (Skolem)

from
http://www.earlham.edu/~peters/cours...s/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/cours...s/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.

ISBN 1876347724