Godel did not destroy the Hilbert Frege Russell programme

The Australian philosopher colin leslie dean shows that

Godel did not destroy the Hilbert Frege Russell programme to create a
unitary deductive system in which all mathematical truths can can be
deduced from a handful of axioms

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

Godel is said to have shattered this programme in his paper called "On
formally undecidable propositions of Principia Mathematica and related
systems"

but this paper it turns out had nothing to do with Principia Mathematica
and related systems" but instead with a completly artificial system
called P Godel uses axioms which where not in his version of PM thus his
proof/theorem cannot apply to PM thus he cannot have destroyed the
Hilbert Frege Russell programme and also his system P is artificial and
applies to no system anyways

colin leslie dean shows that Godel constructs an artificial system P made
up of Peano axioms and axioms including the axiom of reducibility-
which is not in the edition of PM he says he is is using. This system
is
invalid as it uses the invalid axiom of reducibility. Godels theorem has
no value out side of his system P and system P is invalid as it uses the
invalid axiom of reducibility


godel uses the axiom of reducibility
he tell us he is going to use it

â??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)â??

NOTE HE SAYS HE IS USEING 2ND ED PM -where the axiom of reducibility was
repudiated given up and dropped



and he uses it in his axiom 1v and formular 40

Godel uses the axiom of reducibility axiom 1V of his system is the
axiom of reducibility â??As Godel says â??this axiom represents the axiom
of reducibility (comprehension axiom of set theory)â?? (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 â??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

â?? [40. R-Ax(x) â?¡ (â??u,v,y,n)[u, v, y, n <= x & n Var v & (n+1)
Var u
& u Fr y & Form(y) & x = u â??x {v Gen [[R(u)*E(R(v))] Aeq y]}]

x is a formula derived from the axiom-schema IV, 1 by substitution
â??



http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps.


"The system P of footnote 48a is Godelâ??s
streamlined version of Russellâ??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, â??This axiom plays the role of
[Russellâ??s] axiom of reducibility (the comprehension axiom of set
theory).â??


(BUT

IT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL TOOK THE AXIOM
OF REDUCIBILITY OUT OF THAT EDITION â?? which Godel must have known.

The Cambridge History of Philosophy, 1870-1945- page 154

http://books.google.com/books?id=I0...WOzml_RmOLy_JS0
Quote

â??In the Introduction to the second edition of Principia, Russell
repudiated Reducibility as 'clearly not the sort of axiom with which we
can rest content'â?¦Russells own system with out reducibility was
rendered incapable of achieving its own purposeâ??

quote page 14
http://www.helsinki.fi/filosofia/gts/ramsay.pdf.

Russell gave up the Axiom of Reducibility in the second edition of
Principia (1925)â??



http://books.google.com.au/books?id...sh0US6QrI&hl=en
Phenomenology and Logic: The Boston College Lectures on Mathematical
Logic
and Existentialism (Collected Works of Bernard Lonergan) page 43

"in the second edition Whitehead and Russell took the step of using
the simplified theory of types DROPPING THE AXIOM OF REDUCIBILITY and not
worrying to much about the semantical difficulties"




Godels paper is called

ON FORMALLY UNDECIDABLE PROPOSITIONS

OF PRINCIPIA MATHEMATICA AND RELATED

SYSTEMS

but he uses an axiom that was not in PRINCIPIA MATHEMATICA thus his
proof/theorem has nothing to do with PRINCIPIA MATHEMATICA AND RELATED
SYSTEMS at all

Godels proof is about his artificial system P -which is invalid as it
uses
the ad hoc invalid axiom of reducibility


system P is the system from which he derives his incompleteness theorem
quote from the van Heijenoort translation


[quote]â??Theorem XI. Let κ be any recursive consistent63 class of
FORMULAS;
then the SENTENTIAL FORMULA stating that κ is consistent is not
κ-PROVABLE; in particular, the consistency of P is not provable in
P,64provided P is consistent (in the opposite case, of course, every
proposition is provable [in P])". (Brackets in original added by
Gödelâ??to help the readerâ??, translation and typography in van
Heijenoort
1967:614)

Godel tells us
"P is essentially the system which one obtains by building the logic of
PM
around Peanos axioms..."

and
ystem P contain the axiom of reducibility


Godel uses the axiom of reducibility axiom 1V of his system is the
axiom of reducibility â??As Godel says â??this axiom represents the
axiom
of
reducibility (comprehension axiom of set theory)â??

http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps.


"The system P of footnote 48a is Godelâ??s
streamlined version of Russellâ??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, â??This axiom plays the role of [Russellâ??s]
axiom of reducibility (the comprehension axiom of set theory).â??



EVERY ONE KNEW THAT AR WAS NOT IN 2ND ED PM EVEN

GODEL BUT NO ONE SAID
ANYTHING

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