GiantEvil said

colin leslie dean points out godels theorem is invalid-illegitimate because it uses the invalid axiom of reducibilityMr edam421, if Mr. Kurt Godel's incompleteness theorem is wrong, pick an axiom and prove it.

http://www.scribd.com/doc/32970323/G...d-illegitimate

regardless of how faultless godels proof/logic is

his proof is invalid as it uses the invalid axiom of reducibility

godel axiom 1v is the axiom of reducibility

and his formula 40 uses the axiom of reducibility

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)ramsy wittgenstien russel etc say this axiom is invalid“ [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u & u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]

x is a formula derived from the axiom-schema IV, 1 by substitution “(K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965)

godel tells us that in his proof of the incompletness theorem he uses axiom 1v

NOTE HE SAYS PROOF

system p is“In the proofof 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

“P is essentially the system obtained by superimposing on the Peano axioms the logic of PM”

AXIOMS OF P

“I.

Gödel uses only three of the Peano postulates; the others are supplanted by the axion-schemata defined later.

1. ~(Sx1 = 0)

Zero is the successor of no number. Expanded into the basic signs, the axiom is: ~(a2 ∀ (~(a2(x1)) ∨ a2(0)))

This is the smallest axiom in the entire system (although there are smaller theorems, such as 0=0).

2. Sx1 = Sy1 ⊃ x1 = y1

If x+1 = y+1 then x=y. Expanding the ⊃ operator we get: ~(Sx1 = Sy1) ∨ (x1 = y1) And expanding the = operators we get: ~(a2 ∀ (~(a2(Sx1)) ∨ a2(Sy1))) ∨ (a2 ∀ (~(a2(x1)) ∨ a2(y1)))

3. x2(0).x1 ∀ (x2(x1) ⊃ x2(fx1)) ⊃ x1 ∀ (x2(x1))

The principle of mathematical induction: If something is true for x=0, and if you can show that whenever it is true for y it is also true for y+1, then it is true for all whole numbers x.

[178]II. Every formula derived from the following schemata by substitution of any formulae for p, q and r.

1. p ∨ p ⊃ p

2. p ⊃ p ∨ q

3. p ∨ q ⊃ q ∨ p

4. (p ⊃ q) ⊃ (r ∨ p ⊃ r ∨ q)

III. Every formula derived from the two schemata

1. v ∀ (a) ∨ Subst a(v|c)

2. v ∀ (b ⊃ a) ∨ b ⊃ v ∀ (a)

by making the following substitutions for a, v, b, c (and carrying out in I the operation denoted by "Subst"): for a any given formula, for v any variable, for b any formula in which v does not appear free, for c a sign of the same type as v, provided that c contains no variable which is bound in a at a place where v is free.23

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).

V. Every formula derived from the following by type-lift (and this formula itself):

1. x1 ∀ (x2(x1) ≡ y2(x1)) ∨ x2 = y2.

This axiom states that a class is completely determined by its elements.”

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

"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).”

NOW RAMSEY RUSSELL WITGENSTIEN say the axiom of reducibility is invalid

“As a corollary, the axiom of reducibility was banished as irrelevant to mathematics ... The axiom has been regarded as re-instating the semantic paradoxes” - http://mind.oxfordjournals.org/cgi/r...07/428/823.pdfalso godel used the second edition of PM4) Russell Ramsey and Wittgenstein regarded it as illegitimate Russell abandoned this axiom – in 2nd ed PM- and many believe it is illegitimate and must be not used in mathematics

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

http://plato.stanford.edu/entries/pr...a-mathematica/

“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

http://books.google.com/books?id=gDz...SvhA#PPA119,M1

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

but in that edition russell repudiated the axiom- so godels proof uses a version of PM that russell abandoned

“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 reducibilityIT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL ABANDONED REJECTED GAVE UP DROPPED THE AXIOM OF REDUCIBILITY IN THAT EDITION – which Godel must have known. Godel used a text in PM that based on Russells revised version of PM in 2nd ed PM Russell had rejected abandoned dropped as stated in the introduction. Godel used a text with the axiom of reducibility in it but Russell had abandoned rejected dropped this axiom as stated in the introduction. Godel used a rejected text as it used the rejected axiom of reducibility.

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

http://books.google.com/books?id=I09...Ozml_RmOLy_JS0

Quote

“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'…Russells own system with out reducibility was rendered incapable of achieving its own purpose”

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)”

Phenomenology and Logic: The Boston College Lectures on Mathematical Logic and Existentialism (Collected Works of Bernard Lonergan) page 43

http://books.google.com.au/books?id=...h0US6QrI&hl=en

“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”

In Godels collected works vol 11 page 133

http://books.google.com.au/books?id=...-iLznOYs&hl=en

it says AR is dropped

quote

In the second edition of Principia (at least in the introduction) ...the axiom of reducibility is

dropped