System P is Godels meta theory not his object theory

Many argue that Godels system P is his object theory and that he only uses
PM as an example of his incompleteness theorem. This is wrong
The Australian philosopher colin leslie dean points out that system P is
Godels meta-theory and not his object theory as it is based upon system P
that Godel derives his general incompleteness theorem and that PM is part
of godels meta theory and not an example of his incompleteness theorem

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

[QUOTE

That P is the meta theory is clearly seen when Godels gives us his general
proof of undecidability which uses P



He states


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

]Proposition VI: To every Ï?-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).
[/QUOTE]

[QUOTE]

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

Proposition VI: To every Ï?-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).

Proof: Let c be any given recursive Ï?-consistent class of formulae. We
define:

Bwc(x) â?¡ (n)[n <= l(x) â?? Ax(n Gl x) â?¨ (n Gl x) ε c â?¨
(Ep,q){0 < p,q < n & Fl(n Gl x, p Gl x, q Gl x)}] & l(x) > 0 (5)

(cf. the analogous concept 44)

x Bc y â?¡ Bwc(x) & [l(x)] Gl x = y (6)

Bewc(x) â?¡ (â??y)y Bc x (6.1)

(cf. the analogous concepts 45, 46)

Etc

Etc


"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
[191]a system made by adding a recursively definable Ï?-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.â??. (ibid, p.28)


CLEARLY GODEL IS MAKING SWEEPING CLAIMS JUST BASED UPON HIS P PROOF
Clearly P is part of the meta- theory. 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

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