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Science Forum Index » Philosophy - Meta Forum » Godels incompleteness theorem proven invalid
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| elsiemelsi |
 Posted: Fri Oct 12, 2007 10:31 pm |
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The Australian philosopher Colin Leslie Dean argues Godels incompleteness
theorems are invalid because the axioms he uses are invalid and there is
paradox in his system. His theorems may be an example of brilliant logic
but philosophically his theorems are invalid as they have invalid starting
premises
http://gamahucherpress.yellowgum.com/books/philosophy/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)
PARADOX
There is a simple paradox in Godel's incompleteness theorem, pointed out
by the Australian philosopher Colin Leslie Dean, that invalidates it.
Godel claims that:
:'''undecidability "does not depend upon the special nature of the
constructed systems but rather holds for a very wide class of formal
systems"'''
and yet in his proof he states undecidability is dependent on formal
system P '''"hence in every formal system which satisfies assumptions 1
and 2 [ from system P] and is w - consistent there exist undecidable
propositions"'''. Thus Godel's incompleteness theorem ends in paradox and
is meaningless or invalid.
Godel makes the claim that there are undecidable propositions in a formal
system that don't depend upon the special nature of the formal system.
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
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).
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. :
:"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. Thus, we have a dilemma:
EITHER
:1) 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 Godel's incompleteness
theorem is invalid and a complete failure.
Thus either way Godel's incompleteness theorem are invalid and a complete
failure :either due to the paradox in his theorem or the invalidity of his
axioms."
INVALID AXIOMS
Godel proved his incompleteness theorem 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
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
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
“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)
But both these axioms have major problems. The axiom of reducibility
mathematicians say must be banished from mathematics
AXIOM OF REDUCIBILITY
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
“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
“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 )
IMPREDICATIVE DEFINITIONS
Godel uses impredicative definitions: As he states “ 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.)
Yet Godels has argued that impredicative definitions destroy mathematics
and make it false. As he states "consider this rather as a proof that the
vicious circle principle is false than that classical mathematics is
false”. http://www.friesian.com/goedel/chap-1.htm
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. 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.
“http://en.wikipedia.org/wiki/Preintuitionism
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/AxiomOfReducibility.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/AxiomOfReducibility.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/AxiomOfReducibility.html) |
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| ccaabboooossee@gmail.com |
| Posted: Wed Oct 31, 2007 3:02 pm |
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Guest
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" 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)"
With that many qualifications you just have to take this guy
seriously. Right? Right?
But really, with so many spelling mistakes and ungrammatical sentences
concentrated on the first page I didn't even bother to read further.
Dear Mr Dean. Learn to proof-read and then maybe you will be taken
seriously. |
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