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Aedes

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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

Reply
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 proofof Proposition VI the only properties of thesystem P EMPLOYEDwere the following:

1.The class of axiomsand 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

Reply
Tue 20 May, 2008 09:30 pm

@pam69ur,

pam69ur wrote:

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. hey

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

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

Reply
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 (

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:

"Thesystem Pof footnote 48a is Godel's

streamlined version of Russell's theory of typesbuilt on the natural

numbers as individuals, the system used in [1931]. The last sentence ofthe footnote allstomindtheotherreferencetosettheoryinthatpaper;

KurtGodel[1931,p. 178] wrote of his comprehensionaxiom 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

pam69ur

Reply
Tue 20 May, 2008 10:49 pm

@Arjen,

Quote:

i already have

cant tell us what true statements are

uses an invalid axiom

uses impredicative statements

uses impredicative axiom

falls into 3 paradoxes

where is your trouble

have you read deans book-perhaps that might help

Aedes

Reply
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

Reply
Wed 21 May, 2008 12:31 am

@Aedes,

Quote:

the axiom of reducibility is a mathematical axiom He can't explain it in plain English, because it is not a philosophical or logical proof -- it's a mathematical proof,

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

Reply
Wed 21 May, 2008 12:48 am

@pam69ur,

pam69ur wrote:

That is such unbelievable nonsense. Godel doesn't the axiom of reducibility is a mathematical axiom

and is mathematically invalid

thus making godels proof invalid- mathematically

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

Reply
Wed 21 May, 2008 02:59 am

@Aedes,

Quote:

That is such unbelievable nonsense. Godel doesn'tsystem P or AR as an integral part of a proofuse

to repeat godels own words

Quote:

In the proofof Proposition VI the only properties of thesystem P EMPLOYEDwere the following:

1.The class of axiomsand 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:

Thesystem Pof footnote 48a is Godel's

streamlined version of Russell's theory of typesbuilt on the natural

numbers as individuals, the system used in [1931]. The last sentence ofthe footnote allstomindtheotherreferencetosettheoryinthatpaper;

KurtGodel[1931,p. 178] wrote of his comprehensionaxiom 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

Reply
Wed 21 May, 2008 09:36 am

@pam69ur,

Yes, but you selectively leave out the whole premise of proposition VI:Quote:

And immediately after the part you quote:If, instead of w-consistency, mere consistency as such is assumed forc, 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 that17 Gen ris notc-provable, only the consistency ofcis employed (cf. [189]) and from~Bewc(17 Gen r)it follows, according to (15), that for every numberx,Sb(r 17|z(x))isc-provable, and hence thatSb(r 17|Z(x))is notc-provablefor any number.

By addingNeg(17 Gen r)toc, we obtain a consistent but not w-consistent class offormulaec'.c'is consistent, since otherwise17 Gen rwould bec-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 soa 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 classcconsists of a finite number offormulae(with or without those derived therefrom bytype-lift). Every finite classais naturally recursive. Letabe the largest number contained ina. Then in this case the following holds forc:

[CENTER][/CENTER]

cis 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.

Quote:

and in proposition VI itself he also says this:

Hence in every formal system that satisfies assumptions 1 and 2 and is w-consistent, undecidable propositions exist of the form(x) F(x), whereFis 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 ofP) are derived by substitution from a finite number of schemata.48a

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: Toeveryw-consistent recursive classcofformulaethere correspond recursiveclass-signsr, such that neitherv Gen rnorNeg (v Gen r)belongs toFlg(c)(wherevis thefree variableofr).

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

Quote:

1.17 Gen ris notc-provable

2.Neg(17 Gen r)is notc-provable.

Neg(17 Gen r)is therefore undecidable inc, so that Proposition VI is proved.

Godel is

pam69ur

Reply
Wed 21 May, 2008 05:18 pm

@Aedes,

Quote:

assumption 1 and 2 are from PHence in every formal system that satisfies assumptions 1 and 2 and is w-consistent, undecidable propositions exist of the form(x) F(x), whereFis 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.

Quote:

you point outIn the proof of Proposition VI the only properties of thesystem Pemployed were the following:

1.The class of axiomsand 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 definablein 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

[191]a system made by adding a

Quote:

but what follows is- as i have shown The general result as to the existence of undecidable propositions reads:

Proposition VI: Toeveryw-consistent recursive classcofformulaethere correspond recursiveclass-signsr, such that neitherv Gen rnorNeg (v Gen r)belongs toFlg(c)(wherevis thefree variableofr).

Quote:

In the proof of Proposition VI the only properties of thesystem Pemployed were the following:

1.The class of axiomsand 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 definablein the system P(in the sense of Proposition V).

you say

Quote:

1.17 Gen ris notc-provable

2.Neg(17 Gen r)is notc-provable.

Neg(17 Gen r)is therefore undecidable inc, so that Proposition VI is proved.

in what follows is -as i have shown

Quote:

In the proof of Proposition VI the only properties ofthe system P employedwere the following:

1.The class of axiomsand 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 definablein 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

[191]a system made by adding a

Aedes

Reply
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

Reply
Wed 21 May, 2008 08:03 pm

@Aedes,

Quote:

Can you respond to this without resorting to quotations?

all i can say is

go read godels own words

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

Reply
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

Reply
Wed 21 May, 2008 11:22 pm

@Aedes,

Quote:

GODELS PROOF IS ONLY APPLICABLE TO SYSTEM P AND NOT I have read Godel's words, and he makes abundantly clear that he is making a demonstration from within the boundaries of an axiomatic systemthat 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.

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 formulaethere 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 VIthe 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

[191]a system made by adding a recursively definable ω-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

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 propertiesof the system P

employed were the following"

and from that proof he gets his incompleteness theorem AND FROM NO WHERE ELSE

Aedes

Reply
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

Reply
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 formulaethere 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 VIthe 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 theonly 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 2and 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

Reply
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

Reply
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!!!!

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