# Mathematicians are in deep trouble for 2 reasons

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3. » Mathematicians are in deep trouble for 2 reasons

Sun 11 May, 2008 06:18 pm
The australian philosopher colin leslie dean points out Mathematicians are
in deep trouble for 2 reasons

1) Godel cannot tell us what makes a statement true-thus his incompleteness theorem is meaningless

2 the skolem paradox makes set theory ZFC inconsistent

1)
mathematician have so much invested in godels incompleteness theorem
much maths is reliant on it
but at the time godel wrote his theorem he had no idea of what truth was
as peter smith the Cambridge expert on godel admitts

Quote:

Gödel didn't rely on the notion
of truth
but truth is central to his theorem
as peter smith kindly tellls us

http://assets.cambridge.org/97805218/57840/excerpt/9780521857840_excerpt.pdf

Quote:
you see godel referes to true statement
but Gödel didn't rely on the notion
of truth

now because Gödel didn't rely on the notion
of truth he cant tell us what true statements are
thus his theorem is meaningless

this puts mathematicians in deep trouble because all the modern idea derived
from godels theorem have no epistemological or mathematical worth for we
dont know what true statement are

Gödel's incompleteness theorems - Wikipedia, the free encyclopedia...

Quote:
Gödel's first incompleteness theorem, 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.
without a notion of truth we dont know what makes those statements true
thus the theorem is meaningless

and modern mathematics is in deep trouble for useing a meaningless theorem

2) skolem discovered a paradox which makes set theory inconsistent

Skolem's paradox - Wikipedia, the free encyclopedia

Quote:
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
freankel and most mathematicians at the time saw it as an antinomy contradiction

http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Quote:
Neither have the books yet been closed on the antinomy, nor has
agreementon its significance and possible solution yet been reached." ""
(Abraham
Fraenkel)

and that

"most mathematicians followed fraenkels skepticiam

the wiki article says it appears to be a contradiction based on the skolem attempted solution

in regard to skolem relativism attempt at resolution - which is at
present is not accepted

skolems relativistic solution had the affect of destroying set theory -of which he himself noted
http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps

Quote:
"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." - ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147
Peter Suber points out the problem with Skolems relativism IE IT DESTROYS SET THEORY

http://www.earlham.edu/~peters/cours...skol.htm#amb3]

Quote:
This means that there simply are no sets whose cardinality is absolutely uncountable. For many, this view guts set theory, arithmetic, and analysis. It is also clearly incompatible with mathematical Platonism which holds that the real numbers exist, and are really uncountable, independently of what can be proved about them.

(John von Neumanns states

Quote:
"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being noway of rehabilitating this theory is known."
of which a few mathematician also agreed]

now rather than solving the paradox before moving on with set theory
mathematicians just ignored it and used set theory for all sorts of
proofs

Now mathematicians are in deep trouble for there is now so much invested in
set theory that the skolem paradox threatens the very foundations of
mathematics

so some mathematician now try to argue away the paradox by saying it is
but
skolems paradox want go away it is at present unable to be disproved
and modern maths is buried so much in trouble for useing set theory they cant
get out

so we have
either the paradox means set theory ZFC is inconsistent
or
set theory is destroyed

Arjen

Mon 12 May, 2008 02:04 am
@pam69ur,
Say pam69ur, why are you repeating these complaints in every topic you open, but not arguing them anywhere?

pam69ur

Mon 12 May, 2008 05:15 am
@Arjen,
Quote:
Say pam69ur, why are you repeating these complaints in every topic you open, but not arguing them anywhere?

surely the evidence presented is argument enough
one dose not have to wrie 20000 word when a few quote do it

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

thus this is meaningless
Quote:
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.

you dont need 200000 words of argument

similarly you dont need 200000 words of argument

Quote:
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

to see that set theory ZFC is inconsistent

Arjen

Mon 12 May, 2008 05:53 am
@pam69ur,
Is not the presumption of thruth of one's own arguments (as opposed to the study thereof) a strange way of discussing things?

pam69ur

Mon 12 May, 2008 06:14 am
@Arjen,
Quote:
Is not the presumption of thruth of one's own arguments (as opposed to the study thereof) a strange way of discussing things?

if you disagree with the claims feel free to give your 200000 word rebutale

Arjen

Mon 12 May, 2008 07:45 am
@pam69ur,
I have done so in a number of your topics, with far less than 20000 words I might add, but you never argue your point. You merele state it and the move on to another topic to state the same thing. What are you trying to do?

Aedes

Mon 12 May, 2008 12:53 pm
@Arjen,
Arjen,

He's gone all over the entire internet under the names pam69ur, pam666, nightdreamer, and daymare, saying the same thing repeatedly on about a million different forums. He's either Colin Leslie Dean himself or he's Dean's press agent.

Interestingly I did a journal article search for Colin Leslie Dean in JSTOR, which is one of the largest academic journal search engines, and did not find a single publication.

And the irony is that such a huge contribution to mathematics would certainly be heralded in academic circles, not on net forums for laypeople.

Finally, mathematics is applied. I don't think mathematicians will wake up feeling any differently irrespective of what people think about Godel's theorems. Math will still send rockets to the space station and engineer new cars.

Arjen

Mon 12 May, 2008 01:07 pm
@Aedes,
I was not aware of these facts, thank you Aedes. Although not being published does not mean not brilliant or not right and I was interested in the discussion, but the information you provided does shine a totally different light on the matter.

Aedes

Mon 12 May, 2008 01:14 pm
@Arjen,
Arjen wrote:
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), I'd think there would be a great deal of interest in this subject, which makes one wonder why Dean's only audience seems to be internet forums.

And also I'd think that overturning Godel would save math, not threaten it, because didn't Godel disrupt the preconception that math was a logically intact system?

Arjen

Mon 12 May, 2008 01:20 pm
@Aedes,

Aedes

Mon 12 May, 2008 01:31 pm
@Arjen,
I've read some synopses of the language and logic philosophy of the late 19th and early 20th century, and one of the major movements was to simplify logic to remove lots of the vagaries of language from it. The major players in this were Frege, Russell, and Wittgenstein. The output of this was symbolic logic.

Bertrand Russell took it upon himself to assert that all of mathematics could be expressed through logic, and he set out to prove this (though he didn't exactly make it to complex, higher order math in his venture). But Godel, by showing that math was not in fact a tautology, showed that not all of math could be translated directly to logic.

And Russell responded quite despairingly, writing at one point that if this aspect of math could not be expressed logically, then the rest of math could potentially fall apart logically as well.

Russell never worked on this project again (and he lived and worked until he was around 98 years old) and basically gave up logic thanks to Godel.

Now keep in mind I haven't studied this academically, I've only read it in synopses, so I'm sure a well-trained logician would have a more complete view.

Arjen

Mon 12 May, 2008 02:05 pm
@Aedes,
Do you happen to know of any sources?

Aedes

Mon 12 May, 2008 02:22 pm
@Arjen,
Yes, I'll link them later on tonight.

pam69ur

Mon 12 May, 2008 04:37 pm
@Arjen,
Quote:
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)

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

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

for two reasons

1) as pointed out in this thread godel does not tell us what makes a statement true -thus his incompleteness theorem is meaningless

the Hilbert idea was that true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

Quote:
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

but it is said Godel destroyed this with his incompleteness theorem

Quote:
, 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.

but as pointed out in this thread godel does not tell us what makes a statement true -thus his incompleteness theorem is meaningless

2)

Godel is said to have shattered this programme in his paper called "On
formally undecidable propositions of Principia Mathematica and related
systems". But this paper it turns out had nothing to do with Principia Mathematica and related systems" , but instead with a completly artificial system called P Godel uses axioms which where not in his version of PM Thus his proof/theorem cannot apply to PM thus he cannot have destroyed the Hilbert Frege Russell programme and also his system P is artificial and
applies to no system anyways

Colin leslie dean shows that Godel constructs an artificial system P made
up of Peano axioms and axioms including the axiom of reducibility-
which is not in the edition of PM he says he is is using. This system
is invalid as it uses the invalid axiom of reducibility. Godels theorem has
no value out side of his system P and system P is invalid as it uses the
invalid axiom of reducibilitygodel uses the axiom of reducibility
he tell us he is going to use it

Quote:
"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)"
NOTE HE SAYS HE IS USEING 2ND ED PM -where the axiom of reducibility was repudiated given up and droppedand he uses it in his axiom 1v and formular 40

Quote:
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
"

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

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)."
(BUT

IT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL TOOK THE AXIOM OF REDUCIBILITY OUT OF THAT EDITION - which Godel must have known.

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

Quote

Quote:
quote page 14
http://www.helsinki.fi/filosofia/gts/ramsay.pdf.

Quote:
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

Quote:
"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"
Godels paper is called

ON FORMALLY UNDECIDABLE PROPOSITIONS

OF PRINCIPIA MATHEMATICA AND RELATED

SYSTEMS

but he uses an axiom that was not in PRINCIPIA MATHEMATICA thus his
proof/theorem has nothing to do with PRINCIPIA MATHEMATICA AND RELATED SYSTEMS at all

Godels proof is about his artificial system P -which is invalid as it
uses the ad hoc invalid axiom of reducibility system P is the system from which he derives his incompleteness theorem quote from the van Heijenoort translation

Quote:
Godel tells us

Quote:
"P is essentially the system which one obtains by building the logic of PM around Peanos axioms..."
and
system P contain the axiom of reducibility

Godel uses the axiom of reducibility axiom 1V of his system is the
axiom of reducibility "As Godel says

Quote:
"this axiom represents the axiom of
reducibility (comprehension axiom of set theory)"
http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps.

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)."
EVERY ONE KNEW THAT AR WAS NOT IN 2ND ED PM EVEN

GODEL BUT NO ONE SAID
ANYTHING

Aedes

Mon 12 May, 2008 07:55 pm
@pam69ur,
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)

pam69ur wrote:
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.

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.

pam69ur

Mon 12 May, 2008 08:23 pm
@Aedes,
Quote:
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

Aedes

Mon 12 May, 2008 08:33 pm
@pam69ur,
pam69ur wrote:
1)russell knew godel used his invalid axiom AR ramsey knew it
they all knew it and said nothing
That's utter nonsense (as Russell's own correspondance would show you -- his letters are published in the academic literature). Russell was perhaps the most arrogant, obstinate, self-promoting philosopher in the entire 20th century. There is absolutely no way he would be silent about it, because he wasn't silent about anything else. Under no circumstances would Russell quietly lie that Godel had crushed and invalidated his life work (up to that point). And yet that is exactly what Russell said.

Quote:
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"
I don't think that matters to either his hypothesis or his proof, because that was NOT the purpose of his theorem. His theorem was that axioms cannot be proved within their own closed logical system. So thanks for nothing, Colin Leslie, the only thing that's meaningless is this particular critique.

Quote:
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
yes, we know, as you've said (in identical language) in about 26 of your 39 posts here.

And you're also COMPLETELY missing the point. The point is not whether Godel's incompleteness theorem was flawed, flawless, whatever. The REAL question is whether there can be such a thing as a true, mathematically sound incompleteness theorem irrespective of whether Godel was the author or not. Colin Leslie Dean, who is not so secretly all over the net loudly promoting his cute little attack on Godel (rather than a meaningful attack on incompleteness), has not actually addressed the real question. If he's the philosopher he wants everyone to think he is, he can stop obsessing over Godel and actually take on incompleteness instead.

pam69ur

Mon 12 May, 2008 09:16 pm
@Aedes,
Quote:
That's utter nonsense (as Russell's own correspondance would show you
just read the proof dated 1932
it states it uses AR either russell did not read the bloody thing
or he kept quite
just as ramsey either did not read the bloody thing
or kept quite

if what you say is true how do you 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

Quote:
"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.

but blindly ignore what colin leslie dean has pointed out

1) he used an invalid axiom ie axiom of reducibility

Quote:
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:
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
2)constructs impredicative statements

Quote:
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
3)uses peano which is impredicative

Quote:
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.
axiom 5 peano
Quote:
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.
4)he cant tell us what makes statements true

Quote:
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.
but Peter smith the cambridge expert on godels says
Quote:
Quote:
Gödel didn't rely on the notion
of truth
5)he uses a discarded rejected version of PM

Quote:
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

thus on a number of points godels theorem is invalid meaningless and rubbish

Aedes

Mon 12 May, 2008 09:23 pm
@pam69ur,
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.

Quote:

pam69ur

Mon 12 May, 2008 09:47 pm
@Aedes,
Quote:
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

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

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