Godels imcompleteness theorem depends on the axioms of PM- but these axioms are invalid

The Australian philosopher has pointed out that Godels incompleteness
theorem depends upon the axioms of PM -but these axioms are invalid


what clearly shows that the above arguments is valid is that Godel uses
the axioms of P -which are those of PM to make statements about the
undecidability of formal system

extracts from his book-pasted below



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


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

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


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.


see the arguments for this in the following book

The Australian philosopher has pointed out that Godels incompleteness
theorem depends upon the axioms of PM -but these axioms are invalid


what clearly shows that the above arguments is valid is that Godel uses
the axioms of P -which are those of PM to make statements about the
undecidability of formal system

extracts from his book-pasted below

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


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

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


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.


see the arguments for this in the following book



.

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