Say pam69ur, why are you repeating these complaints in every topic you open, but not arguing them anywhere?

Gödel didn't rely on the notion
of truth

For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.

set theory which contains only a countable number of objects. However, it must contain the aforementioned uncountable sets. This appears to be a contradiction, since the uncountable sets are subsets of the (countable) domain of the mode

Is not the presumption of thruth of one's own arguments (as opposed to the study thereof) a strange way of discussing things?

Although not being published does not mean not brilliant or not right.

Of course, and it might be in press or have been presented at meetings. Still, considering that Godel singlehandedly ended the effort to unify logic with math (and threw Bertrand Russell into a depression, and caused Russell to completely change his career)

Quote:

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.
The works of , Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system

, perhaps the single most celebrated result in mathematical logic, states that:
For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

"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)"

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

" [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

"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)."

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

"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"

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

"this axiom represents the axiom of
reducibility (comprehension axiom of set theory)"

"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)."

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

Well, Russell certainly thought he had

1)russell knew godel used his invalid axiom AR ramsey knew it
they all knew it and said nothing

for 76 years every one has been saying godel showed there are true statements which cant be proven- ie destroying hilbert/russell programme but no one has bothered to ask [untill colin leslie dcean] "what did godel means by true statement"

and it turns out he had no bloody idea what makes a statement true thus he cannot have destroyed the hilbert/russell programme
as his theorem is meaningless

That's utter nonsense (as Russell's own correspondance would show you

"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). NOTE HE SAYS HE IS USING 2ND ED PM- WHICH RUSSELL ABANDONED REJECTED GAVE UP DROPPED THE AXIOM OF REDUCIBILITY.

Quote:
"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)" (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.12-13)

Quote:
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

Quote:
Ponicare Russell and philosophers argue these types of definitions are invalid Ponicare Russell point out that they lead to contradictions in mathematics

Quote from Godel
" The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... ((K Godel , On undecidable propositions of formal mathematical systems in The undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes, "it covers ground quite similar to that covered in Godels orgiinal 1931 paper on undecidability," p.39.)


What Godel understood by "propositions which make statements about
themselves"

is the sense Russell defined them to be

'Whatever involves all of a collection must not be one of the collection.'
Put otherwise, if to define a collection of objects one must use the total
collection itself, then the definition is meaningless. This explanation
given by Russell in 1905 was accepted by Poincare' in 1906, who coined the
term impredicative definition, (Kline's "Mathematics: The Loss of
Certainty"

Note Ponicare called these self referencing statements impredicative
definitions

texts books on logic tell us self referencing ,statements (petitio
principii) are invalid

axiom 3 from godels system P
Quote:
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.

Quote:
This is the principle of complete induction, it establishes the property
of induction as necessary to the system. Since Peano's axiom is as
infinite as the natural numbers, it is difficult to prove that the
property of P does belong to any x and also x+1. What one can do is say
that, if after some number n of trails that show a property P conserved in
x and x+1, then we may infer that it will still hold to be true after n+1
trails. But this is itself induction. And hence the argument is a vicious
circle.

truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

Quote:
Gödel didn't rely on the notion
of truth

Quote:
IT 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

Colin -- may I call you that? Here is a letter that Bertrand Russell wrote in 1963. He was personally rocked by Godel's work and it altered his entire career