Mathematicians are in deep trouble for 2 reasons

Aedes

Tue 20 May, 2008 09:08 pm
@pam69ur,
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.

pam69ur

Tue 20 May, 2008 09:22 pm
@Aedes,
Quote:
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.

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

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)

system P has axiom 1v
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).

and ramsey and your stanford and editors of godels book have said it is rubbish
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

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 "

Aedes

Tue 20 May, 2008 09:30 pm
@pam69ur,
pam69ur wrote:
hey
godel tells you he uses axiom reducibility- and read dean showing where godel falls into paradox
Reading Dean only shows where Dean falls into paradox. The first incompleteness theorem does not depend on the soundness of AR to begin with, and it may be sound under the circumstances Godel used anyway according to the Stanford article that you've repeatedly cited. The EXACT same article says that it's valid as Godel used it.

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.

pam69ur

Tue 20 May, 2008 10:05 pm
@Aedes,
Quote:
The EXACT same article says that it's valid as Godel used it.

sorry your quote says godel observed in (1944), a form of AR
godels incompleteness theorem was done in 1931-32 not in 1948

Quote:
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.

godel used russells AR
as the quote tells you

Quote:
"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).

which is rubbish
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

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 "

you say

Quote:
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.

hey perhaps dean migh go after rosser and all the rest as all products of human thinking end in meaninglessness so will they

but what will be left for all those budding phd s who want to0 make aname for themselves -perhaps dean wil just let them have some fun

Arjen

Tue 20 May, 2008 10:35 pm
@pam69ur,

pam69ur

Tue 20 May, 2008 10:49 pm
@Arjen,
Quote:

cant tell us what true statements are
uses an invalid axiom
uses impredicative statements
uses impredicative axiom

have you read deans book-perhaps that might help

Aedes

Wed 21 May, 2008 12:26 am
@Arjen,
Arjen wrote:
He can't explain it in plain English, because it is not a philosophical or logical proof -- it's a mathematical proof, but he is not a mathematician. He doesn't understand it. Neither do I, but then again I'm not the one claiming to have overturned one of the most significant mathematical achievements in history. A mathematician could certainly be equipped to relate the mathematical coherence of the proof in lay terms that you and I can understand. But all he's doing is cutting and pasting and interspersing the word "rubbish", which correct me if I'm wrong is not part of the mathematical lexicon.

pam69ur

Wed 21 May, 2008 12:31 am
@Aedes,
Quote:
He can't explain it in plain English, because it is not a philosophical or logical proof -- it's a mathematical proof,
the axiom of reducibility is a mathematical axiom
and is mathematically invalid
thus making godels proof invalid- mathematically

ramsey and the editors of godels books are mathematicians
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

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 "

godel has no idea what makes a mathematical statement true
thus mathematically his incompleteness theorem is meaningless

godel uses impredicative statements
poincare is a mathematician and he notes these statements are mathematically invalid
thus godels theorem is invalid

Aedes

Wed 21 May, 2008 12:48 am
@pam69ur,
pam69ur wrote:
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. He offers theorems ABOUT system P. That's the most fundamental and rudimentary element of his first theorem and it's shocking you don't understand that. An axiom can only be invalid if it contradicts other axioms used in the same theorem. That's the whole point of Godel's critique, which applies to all formal logic systems that contain axioms.

This is what happened when you actually presented this idea to a mathematician:

Pam69ur, aka elsiemelsi, on the math forum

Although to be fair, I like two replies in this thread better

pam69ur

Wed 21 May, 2008 02:59 am
@Aedes,
Quote:
That is such unbelievable nonsense. Godel doesn't use system P or AR as an integral part of a proof

to repeat godels own words

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)
Quote:
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).

Aedes

Wed 21 May, 2008 09:36 am
@pam69ur,
Yes, but you selectively leave out the whole premise of proposition VI:
Quote:
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.
And immediately after the part you quote:
Quote:

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
and in proposition VI itself he also says this:

Quote:
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).

He gives the proof, then concludes (I've truncated this):

Quote:
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.

Godel is CLEARLY constructing his proof around the ramifications of a certain kind of axiomatic statement as well as the converse of that statement. The validity of that axiom is irrelevant, because it can only be valid in relation to the theorem that contains it. However, its structure is what is relevant, and that's the whole basis of this (and every other) proposition.

pam69ur

Wed 21 May, 2008 05:18 pm
@Aedes,
Quote:
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.
assumption 1 and 2 are from P

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
you point out
Quote:
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).
but what follows is- as i have shown
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).

you say
Quote:
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 what follows is -as i have shown

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

Aedes

Wed 21 May, 2008 07:20 pm
@pam69ur,
Yes, exactly, and what you for some reason are failing to see in your very quotes is how Godel is using system P as an example, not relying on it as an integral part of the proof.

Can you respond to this without resorting to quotations?

pam69ur

Wed 21 May, 2008 08:03 pm
@Aedes,
Quote:
Can you respond to this without resorting to quotations?

all i can say is
godel tells us he employes/uses system P in his proof
the maths department at the uni of california

tell us that system P is godels streamlined version of russlles theory of types
and the axiom of reducibility is part of that system P

Aedes

Wed 21 May, 2008 09:42 pm
@pam69ur,
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.

pam69ur

Wed 21 May, 2008 11:22 pm
@Aedes,
Quote:
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.
GODELS PROOF IS ONLY APPLICABLE TO SYSTEM P AND NOT other axiomatic systems.

dean notes

Quote:
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

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

Aedes

Thu 22 May, 2008 09:23 am
@pam69ur,
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. If you're concerned about its generalizability to other systems, then it's awfully strange that you haven't taken the time to refute the proofs of the first incompleteness theorem as applied by other mathematicians outside of system P. Godel had it right, and even if you don't like his methodology his theorem has been repeatedly upheld using different (and more parsimonious) methods. So even if you succeed in poking a hole in Godel's proof it wouldn't matter because his proof has been redone and improved upon.

pam69ur

Thu 22 May, 2008 09:50 am
@Aedes,
Quote:
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.

sorry system P is not the example
it is the very foundation of his theorem
Quote:
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

he uses system P to prove/derive his theorem
which he cant then generalise to other systems
as it can only be relevent to the system from which it is derived ie system P

he tells you this anyway

he says any system which has the axioms of P will be undecidable
and not every system has those axioms
thus he cant then say

Quote:
To every ω-consistent recursive class c of formulae

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

Aedes

Thu 22 May, 2008 10:01 am
@pam69ur,
It's amazing -- you repeat the same quotes again and again and again, and you completely ignore all of the text in Godel's theorem that surrounds it. But I understand, you have an agenda, so why take the time to read the theorem more accurately?

VideCorSpoon

Thu 22 May, 2008 11:28 am
@Aedes,
Pam69ur,

Wow... that you have been able to take Ctrl-C, Ctrl-V to such an extent that it is mind boggling. Kudos.

Since you are inextricably familiar with Godel and logic in particular... I'm just curious, can you solve this?

(∃x) (Gx) --> (y) (Gy --> Jy) / (x) (Gx --> Jx)

(There are only eight steps, so its not too much trouble as long as you are familiar with the material. It's kinda like calling your bluff, so lets see what cards you have.)

Do it as the Australian philosopher Colin Leslie Dean would have, may his infinite wisdom grace our lowly existence!!!!