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 at bottom of post
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 Phe 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
see 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
“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
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
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)
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
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
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 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]”
Gödel is a complete failure as he ends in utter meaninglessness. His
meaningless/paradoxical result comes directly from using axioms 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; 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.
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)
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
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.
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/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_paradox
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 Löwenheim-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/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.
ISBN 1876347724
