Whatever, every time I show a reason why I find your critique to be wrong you selectively cut and paste the same stuff over and over again. Fine, you're right. Mathematicians are in deep deep trouble.

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)

IV. Every formula derived from the schema
1. (∃u)(v ∀ (u(v) ≡ a))
on substituting for v or u any variables of types n or n + 1 respectively, and for a a formula which does not contain u free. This axiom represents the axiom of reducibility (the axiom of comprehension of set theory).

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY


the standford encyclopdeia of philosophy says of AR

Principia Mathematica (Stanford Encyclopedia of Philosophy)

"many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically"

From Kurt Godels collected works vol 3 p.119

Collected Works - Google Book Search

"the axiom of reducibility is generally regarded as the grossest philosophical expediency "

hey
godel tells you he uses axiom reducibility- and read dean showing where godel falls into paradox

The EXACT same article says that it's valid as Godel used it.

Interestingly, the basic idea underlying Russell's ramified hierarchy of types is a crucial ingredient for Godel's later consistency proof of the continuum hypothesis via its inner model L of constructible sets. Also, as already observed in Godel (1944), a form of AR becomes true in L in the sense that, roughly, an arbitrary propositional function of natural numbers is extensionally equivalent to some function of order alpha, for some countable ordinal alpha.

"The system P of footnote 48a is Godel's
streamlined version of Russell's theory of types built on the natural
numbers as individuals, the system used in [1931]. The last sentence ofthe footnote allstomindtheotherreferencetosettheoryinthatpaper;
KurtGodel[1931,p. 178] wrote of his comprehension axiom IV, foreshadowing
his approach to set theory, "This axiom plays the role of [Russell's]
axiom of reducibility (the comprehension axiom of set theory).

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY


the standford encyclopdeia of philosophy says of AR

Principia Mathematica (Stanford Encyclopedia of Philosophy)

"many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically"

From Kurt Godels collected works vol 3 p.119

Collected Works - Google Book Search

"the axiom of reducibility is generally regarded as the grossest philosophical expediency "

And considering the work that Rosser did confirming and strengthening the First Incompleteness Theorem, I'm shocked that you'd pronounce Godel dead before even taking the time to go after Rosser too.

He can't explain it in plain English, because it is not a philosophical or logical proof -- it's a mathematical proof,

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY


the standford encyclopdeia of philosophy says of AR

Principia Mathematica (Stanford Encyclopedia of Philosophy)

"many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically"

From Kurt Godels collected works vol 3 p.119

Collected Works - Google Book Search

"the axiom of reducibility is generally regarded as the grossest philosophical expediency "

the axiom of reducibility is a mathematical axiom
and is mathematically invalid
thus making godels proof invalid- mathematically

That is such unbelievable nonsense. Godel doesn't use system P or AR as an integral part of a proof

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)

The system P of footnote 48a is Godel's
streamlined version of Russell's theory of types built on the natural
numbers as individuals, the system used in [1931]. The last sentence ofthe footnote allstomindtheotherreferencetosettheoryinthatpaper;
KurtGodel[1931,p. 178] wrote of his comprehension axiom IV, foreshadowing
his approach to set theory, "This axiom plays the role of [Russell's]
axiom of reducibility (the comprehension axiom of set theory).

If, instead of w-consistency, mere consistency as such is assumed for c, then there follows, indeed, not the existence of an undecidable proposition, but rather the existence of a property (r) for which it is possible neither to provide a counter-example nor to prove that it holds for all numbers. For, in proving that 17 Gen r is not c-provable, only the consistency of c is employed (cf. [189]) and from ~Bewc(17 Gen r) it follows, according to (15), that for every number x, Sb(r 17|z(x)) is c-provable, and hence that Sb(r 17|Z(x)) is not c-provable for any number.

By adding Neg(17 Gen r) to c, we obtain a consistent but not w-consistent class of formulae c'. c' is consistent, since otherwise 17 Gen r would be c-provable. c' is not however w-consistent, since in virtue of ~Bewc(17 Gen r) and (15) we have: (x) BewcSb(r 17|Z(x)), and so a fortiori: (x) Bewc'Sb(r 17|Z(x)), and on the other hand, naturally: Bewc'[Neg(17 Gen r)].46

A special case of Proposition VI is that in which the class c consists of a finite number of formulae (with or without those derived therefrom by type-lift). Every finite class a is naturally recursive. Let a be the largest number contained in a. Then in this case the following holds for c:
[CENTER][/CENTER]

c is therefore recursive. This allows one, for example, to conclude that even with the help of the axiom of choice (for all types), or the generalized continuum hypothesis, not all propositions are decidable, it being assumed that these hypotheses are w-consistent.

Hence in every formal system that satisfies assumptions 1 and 2 and is w-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 a system made by adding a recursively definable w-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 and also the axiom system of number theory which consists of the Peano axioms, the operation of recursive definition [according to schema (2)] and the logical rules.48 Assumption 1 is in general satisfied by every system whose rules of inference are the usual ones and whose axioms (like those of P) are derived by substitution from a finite number of schemata.48a

Every w-consistent system is naturally also consistent. The converse, however, is not the case, as will be shown later.

The general result as to the existence of undecidable propositions reads:

Proposition VI: To every w-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).

1. 17 Gen r is not c-provable
2. Neg(17 Gen r) is not c-provable.
Neg(17 Gen r) is therefore undecidable in c, so that Proposition VI is proved.

Hence in every formal system that satisfies assumptions 1 and 2 and is w-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 a system made by adding a recursively definable w-consistent class of axioms.

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

The general result as to the existence of undecidable propositions reads:

Proposition VI: To every w-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).

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).

1. 17 Gen r is not c-provable
2. Neg(17 Gen r) is not c-provable.
Neg(17 Gen r) is therefore undecidable in c, so that Proposition VI is proved.

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

Can you respond to this without resorting to quotations?

I have read Godel's words, and he makes abundantly clear that he is making a demonstration from within the boundaries of an axiomatic system that is applicable to other axiomatic systems. He is not making a case that depends in any way on the internal validity of what he calls system P or on AR. I think it would take a biased and cursory reading to think otherwise.

But our views are on the table, and I don't feel the need to repeat myself.

GODEL INCOMPLETENESS THEOREM IS ONLY APPLICABLE TO THE INVALID SYSTEM P- HE INCORRECTLY GENERALISES IT TO OTHER SYSTEMS

Godels system P is not his object theory but is his main theory from which he derives his incompleteness theorem

godels incompleteness theorem reads- note it says to every ω-consistent
recursive class c of formulae



Godel's first Incompleteness Proof at MROB at MROB



Proposition VI: To every ω-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).

now
1) he derives his incompleteness theorem from system P which is made up of
peano and PM but decietfully says it applyies to other system



quote

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

note his theorem says

to every ω-consistent recursive class c of formulae



but he has only proved his theorem for system P ie PM
so he cant extend that to to every ω-consistent recursive class c of
formulae

"In the proof of Proposition VI the only properties of the system P
employed were the following"
and from that proof he gets his incompleteness theorem AND FROM NO WHERE ELSE

Godel openly generalizes it to other systems (and this has been confirmed repeatedly in other systems since then) and he only uses system P as an example.

Proposition VI: To every ω-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).

In the proof of Proposition VI the only properties of the system P
employed were the following

To every ω-consistent recursive class c of formulae

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