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Science Forum Index » Logic Forum » 3rd paradox in godels incompleteness theorem
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| elsiemelsi |
 Posted: Mon Apr 21, 2008 5:17 am |
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the australian philosopher points out the 3rd paradox in godels
incompleteness theorem
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
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 . Note
from above the version of PM he is using AR was abandoned rejected given
up DROPPED So system P is completely artificial and invalid as it uses the
invalid axiom of reducibility. Thus his theorem has no value outside this
invalid artificial system P
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
--
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