Mathematicians are in deep trouble for 2 reasons

@pam69ur,

pam69ur wrote:

i dont doubt that
but
explain russell keeping quite when godels tells the reader he is using russels rejected 2nd PM with the abandoned invalid axiom of reducibility -which russel read and kept quite or he did not read it

Or maybe he didn't think that it invalidated the theorem.

@Aedes,

Quote:

Or maybe he didn't think that it invalidated the theorem.

but it did
even russell rejected the axiom of reducibility
ramsey said it was rubbish

so they must have know by useing it it made the theorem invalid

note

even though godel calls his paper

Quote:

ON FORMALLY UNDECIDABLE PROPOSITIONS

OF PRINCIPIA MATHEMATICA AND RELATED

SYSTEMS

so one could assume it was valid for godel to do a proof in regard to the non rejected version of PM thus he could use AR

but his incompleteness proof which uses PM IS NOT JUST ABOUT PM

BUT
about every ω-consistent recursive class κ of FORMULAS

Quote:

Theorem VI. For every ω-consistent recursive class κ of FORMULAS there are recursive CLASS SIGNS r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(κ) (where v is the FREE VARIABLE of r).2 (van Heijenoort translation and typsetting 1967:607. "Flg" is from "Folgerungsmenge = set of consequences" and "Gen" is from "Generalisation = generalization" (cf Meltzer and Braithwaite 1962, 1992 edition:33-34) )

and that is why his incompleteness proof is invalid by generalising a proof useing AR to a wide range of systems
his theorem is invalid

in other words godels incompleteness theorem is ment to apply to such systems as ZFC Q etc all based on A REJECTED PM and the invalid axiom AR
if godel had just stuck with non rejected version of PM that would have been find -and completly trivial- but he extended his theorem well beyound PM thus making his theorem invalid

russell either did not read the bloody thing or he kept quite

@Aedes,

Aedes wrote:

Arjen,

Here are some journal references. I have all these essays in PDF form. The first one is a great short synopsis that appearad in the journal Science.

1.
Keith Devlin Science, New Series, Vol. 298, No. 5600. (Dec. 6, 2002), pp. 1899-1900.

2.
John W. Dawson, Jr. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1984, Volume Two: Symposia and Invited Papers. (1984), pp. 253-271.

3. What the
Hugh Lacey; Geoffrey Joseph Mind, New Series, Vol. 77, No. 305. (Jan., 1968), pp. 77-83

4.
David Wayne Thomas PMLA, Vol. 110, No. 2. (Mar., 1995), pp. 248-261.

5.
John W. Dawson, Jr.; Cheryl A. Dawson The Bulletin of Symbolic Logic, Vol. 11, No. 2. (Jun., 2005), pp. 150-171.

And then here are some good links. The Stanford site is far better than Wikipedia for philosophy resources on the web.

Kurt Gödel (Stanford Encyclopedia of Philosophy)

Bertrand Russell (Stanford Encyclopedia of Philosophy)

Well, Russell certainly thought he had.

Nonetheless, I'm sure you'll look forward to some critical response to Colin Leslie Dean's essay if it gains readership in academic circles. I wouldn't be so quick to assume that Dean's own essay is flawless, considering Godel's theorem has been the subject of inordinate study for a good 75 years. I find it more likely that Dean's essay has a hole, at least until it gets its own share of critical review.

Can you get those to me?

@Arjen,

I sent you three of them, the other two are on my other computer, but if I get a chance I'll download them again and send them.

@Arjen,

An interesting read regarding some fundamental problems with mathematics;
This is about Self-Reference

@nameless,

Quote:

An interesting read regarding some fundamental problems with mathematics;
This is about Self-Reference

bear in mind as colin leslie dean has shown godel uses self-referencing statements thus his theorem is invalid

text books on logic say such statements are invalid

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

i will give you something to think about
a lot of maths is built on impredicative statements
but
text books on logic say they are invalid
thus
a lot of maths is logically just rubbish

lets take natural numbers
ie peanos axioms are impredicative

Preintuitionism - Wikipedia, the free encyclopedia

his axiom 5

Quote:

Now godels axiom 3 in his system P is this very axiom -pointed out by dean- again making godels theorem invalid- as colin leslie dean argues

http://www.mrob.com/pub/math/goedel.html

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.

@pam69ur,

pam69ur wrote:

text books on logic say...

You seem to have quite the personal ax to grind, from all the posts that you have made on the subject.
Personally, I never let some academic 'book of rules' hinder my (creative) critical thought processes. I will, perhaps, add it to my data bank.
Human understanding will never progress if everyone must check to make sure that every thought complies with some rule book's list of accepted thoughts.
My guess is that you didn't even look at the link that I so generously posted...

******************
"This is a self-referential sentence."
Is the statement true or false?
And, if true, then other self-referrential statements can be just as true. No?

@pam69ur,

pam69ur wrote:

lets take natural numbers
ie peanos axioms are impredicative

Preintuitionism - Wikipedia, the free encyclopedia

Do you realise that preintuition contains peano's axioms already? If the one is invalid, the other is as well. If the one is invalid and the other valid a paradox would be created in your own reasoning, thus proving the incompleteness in your reasoning. In mean this in the sense that the numbers are paradoxes in preintuition.

What say you to this?

@Arjen,

Quote:

Do you realise that preintuition contains peano's axioms already? If the one is invalid, the other is as well. If the one is invalid and the other valid a paradox would be created in your own reasoning, thus proving the incompleteness in your reasoning. In mean this in the sense that the numbers are paradoxes in preintuition.

Quote:

If the one is invalid, the other is as well.

i would agree

as text books on logic would say

@pam69ur,

Quote:

"This is a self-referential sentence."
Is the statement true or false?
And, if true, then other self-referrential statements can be just as true. No?

"this" refers to what

does it refer to something of a higher type

or does it refer to something of the same type
in which case it is meaningless

you are the one who has not read your own reference

Quote:

They introduced into mathematics a strict system of hierarchical types where a set of one type was only allowed to refer to sets of lower types. The lowest order could only contain "objects" and no sets. The next type could contain only objects and the lowest order of sets. Thus no set could contain itself. Set S would be forbidden. In theory, this also dealt with the two step Liar paradox in that neither (or each) sentence is of a higher type than the other, so it must be meaningless (ie; together they cannot be formulated within the hierarchical system)

but your article says godel delt the death blow to russells plan

but
this takes us back to the thread topic

apart from godel making self referential statements godel cant tell us what true statements are thus his theorem is meaningless -thus it cant have dealt a death blow to russell

Quote:

Because G is a true statement, it is unprovable, therefore, system X is incomplete; it is not powerful enough to capture all truths.

@pam69ur,

pam69ur wrote:

i would agree

as text books on logic would say

@Arjen,

Colin Leslie,

Have you yet refuted the other incompleteness proofs that have been presented by others? Like by Kleene, Rosser, and Kalmar?

Because I think you're really missing the point in attacking Godel's proof as if that is the final word on incompleteness. The real issue is can an incompleteness theorem be true, irrespective of the author. If others have tight incompleteness theorems that you are unable to refute, then your attacks on Godel are disingenuous -- all you can say is that his theorems were imperfect but his premise and conclusions were irrefutable.

@Aedes,

Quote:

colin leslie dean does not use preintuition
he only qoute a site dealing with preintuition
the fact that peanos axiom are impredicaive is independent of the theory of preintuition

the point is the example of peano being impredicative just happened to be meantioned on a preintuition site

@pam69ur,

Quote:

Because I think you're really missing the point in attacking Godel's proof as if that is the final word on incompleteness. The real issue is can an incompleteness theorem be true, irrespective of the author. If others have tight incompleteness theorems that you are unable to refute, then your attacks on Godel are disingenuous -- all you can say is that his theorems were imperfect but his premise and conclusions were irrefutable.

firstly
dean is useing godel as case study to show the meaninglessness of all products of human thinking

secondly
dean would say upon carefull examination the incompletenes proofs of any one Like by Kleene, Rosser, and Kalmar?will end in meaninglessness also

thirdly
1)the premises of godel are not irrefutable
he uses the axiom of reducibility in his incompleteness proof as part of his system P and AR is invalid-thus his theorem is invalid

2) the conclusion of godel ie his theorem is a misapplication of what he did prove
godels proof is only relevant to his system P and not other systems
godel decieves us when he says his theorem is about other systems as his proof is only about system P thus his theorem has validity only to system P - which is invalid due to his invalid axiom AR-
and not to To every ω-consistent recursive class c of formulae

http://www.mrob.com/pub/math/goedel.html

Quote:

The general result as to the existence of undecidable propositions reads:
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).
Proof: Let c be any given recursive ω-consistent class of formulae. We define:

etc
etc

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).

@pam69ur,

In other words, Colin Leslie, you're saying that you have not been able to refute the OTHER proofs of incompleteness by OTHER scholars? Why do you think that is? And what do you think it would mean if you found yourself unable to refute them?

Church, Turing, Kalmar, Kleene, and Rosser and others have all either proved or strongly supported incompleteness. How have you addressed their particular works?

Would you please address this question?

@pam69ur,

pam69ur wrote:

"this" refers to what

Is it not obvious to all that the 'this' refers to the sentence quoted?

Quote:

does it refer to something of a higher type

I do not deal in such artificial distinctions. It is a sentence, a statement, an intelligible and 'meaningful' group of words that will, like any such group, have meaning to some, and neaningless to others.

Quote:

..or does it refer to something of the same type
in which case it is meaningless

It is self referential, and is meaningful to some, as I have said....
"All statements are true in some sense, false in some sense, meaningless in some sense, true and false in some sense, true and meaningless in some sense, false and meaningless in some sense, and true and false and meaningless in some sense." -Robert Anton Wilson
Depends on the 'consciousness' of the observer, I guess...

Quote:

apart from godel making self referential statements godel cant tell us what true statements are thus his theorem is meaningless -thus it cant have dealt a death blow to russell

Reference Wilson's quote again. YOU find meaning where you will. Completely subjective. If you choose to reject a statement because it doesn't meet your criteria for meaning doesn't mean that it is in error, but that you have your perspective. Many, many quite intelligent people have no trouble with the 'truth' of Godel's statement, nor does experiment. You seem to be 'choking on a gnat' that is preventing you from greater perspective/understanding. Perhaps a mental Heimlich maneuver...? *__-

"This sentence has five words."
If you are going to argue the 'truth' ("in some sense") of this self referential sentence, there would seem to be an error in your mental program.

@nameless,

Quote:

n other words, Colin Leslie, you're saying that you have not been able to refute the OTHER proofs of incompleteness by OTHER scholars?

it is not the fact that dean cant refute the others but that colin leslie dean has not bothered at the moment to refute the other proofs
but
would argue that being products of human thinking will end in meaninglessness

you could i suspect contact him via his publisher and drop the guantlet to him

Gamahucher Press Catalogue

or
just think of the gold mine for budding PHd students to do what dean did with godel to

Quote:

Church, Turing, Kalmar, Kleene, and Rosser and others

@pam69ur,

Quote:

Many, many quite intelligent people have no trouble with the 'truth' of Godel's statement,

you must ask yourself just what these quite intelligent people where reading
as the thread has shown you
godel cant tell us what truth is or what makes a statement true
thus his theorem is meaningless babble

for 76 years that have all seen talk of godel showing there are true statements which cant be proven

or this from nameless's source

This is about Self-Reference

Quote:

G is a true statement, it is unprovable, therefore, system X is incomplete; it is not powerful enough to capture all truths.

with out ever asking
oh what does godel mean by truth
if they had of they would have seen he had no idea
and thus his theorem is just meaningless babbel

these quite intelligent people obviously never saw godels invalid axiom 1v ie axiom of reducibility which he uses in system P which is used to proove his theorem

these quite intelligent people never read godels words when he tell them he is constructing impredicative statement -which are invalid

it makes you wonder just what these quite intelligent people read at all or where they just backslapping each other in the 76 year riegn of the standard view orthodoxity

moral is

not believe everything quite intelligent people tell you
ie just look at the egg on hawkins and penrose face now after helluling godel with buckets load of praise with colin leslie deans death blow to a 76 year tradition of worshiping godel

@pam69ur,

pam69ur wrote:

it is not the fact that dean cant refute the others but that colin leslie dean has not bothered at the moment to refute the other proofs

Forgive me if I'm not impressed then, since most of the others I mentioned are generally regarded as improvements on Godel's work. And let me ask for about the 5th time, why has this breakthrough by Dean gotten no attention in the academic world? What kind of response has it gotten?

And don't you find it callous, arrogant, and wholly disrespectful of an academic tradition to repeatedly refer to Godel with pejorative statements like "death blow" and "meaningless"? Certainly no aristotelian would be that disrespectful of a platonist. It's poor form and it lacks academic collegiality.

Speaking of self-referential statements, how do you feel about people anonymously referring to themselves in the third person?

@Aedes,

Quote:

Forgive me if I'm not impressed then, since most of the others I mentioned are generally regarded as improvements on Godel's work. And let me ask for about the 5th time, why has this breakthrough by Dean gotten no attention in the academic world? What kind of response has it gotten?

simple colin leslie dean has not presented it to the academic world
no journals
no nothing
he is a no body who preferes to step out side the grove
he believes that the academic world has and should not have a monopoly on truth

he stands up for the no bodies
truth is truth regardless of whether it is published in a academic journal or a comic book
he is for democracy and spreads his view via the net
he is published by gamahucher press- look up what gamahucher means

i think he would destest the elitism pomposity of the academic world and would like to see it humiliated by a no body destroying one of their gods without using their forms of elitist publications

dean i think is a reader of marcus "one dimensional man " and foucault
who argues
it is not about "truth" but who has the power to tell us what "truth" is

and colin leslie dean is trying to humiliate and over turn the current power centres in regard to truth by presenting argument in the public space and letting people make up their own mind

if you only believe godel from reading others views with out coming to YOUR own views then you worship authority more than you do independent thinking

if you cant come to YOUR OWN view about godel from reading deans arguments but require others to tell you what is to be the accepted view then go read some
marcus
the frankfurt school of social philosophy
and foucault

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Mathematicians are in deep trouble for 2 reasons

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