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Attempts to Refute Cantor's Uncountability Proof?
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Gene Ward Smith
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Joined: 08 Jul 2005
Posts: 409

PostPosted: Sat Jul 08, 2006 8:11 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:

Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing.

You find mathematics as a whole unconvincing, so I find your lack of
conviction to be unconvincing.
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Patrick1182
science forum addict


Joined: 01 Feb 2006
Posts: 55

PostPosted: Sat Jul 08, 2006 8:22 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from which
one might attempt to refute his argument. But before I spend a lot to time
trying to formulate my own argument, it seems reasonable to seek prior art.
Can anybody suggest a source which examines this topic?


http://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps
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Hatto von Aquitanien
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Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 9:11 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Gene Ward Smith wrote:


Quote:
You find mathematics as a whole unconvincing,

Yours is among the most moronic statements I have read in this newsgroup,
and you have some tough competition.


Quote:
so I find your lack of conviction to be unconvincing.

Sorry if I don't accept the holy Aleph-Omega of Mathematics upon the dogma
of others.
--
Nil conscire sibi
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 9:15 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Patrick wrote:

Quote:
Hatto von Aquitanien wrote:
I'm interested to know what attempts have been made to refute Cantor's
proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from
which
one might attempt to refute his argument. But before I spend a lot to
time trying to formulate my own argument, it seems reasonable to seek
prior art. Can anybody suggest a source which examines this topic?


http://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps

I was thinking more in terms of Frege. But I suspect Frege would not have
taking the approach I am likely to take.

"...classical logic was abstracted from the mathematics of finite sets and
their subsets...Forgetful of this limited origin, one afterwards mistook
that logic for something above and prior to all mathematics, and finally
applied it, without justification, to the mathematics of infinite sets.
This is the Fall and original sin of [Cantor's] set theory ..." (Weyl,
1946)
--
Nil conscire sibi
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Sat Jul 08, 2006 9:23 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
Gene Ward Smith wrote:


You find mathematics as a whole unconvincing,

Yours is among the most moronic statements I have read in this newsgroup,
and you have some tough competition.

You've made dumb remarks (as well as remarks both rude and stupid, like
the above) all too often, so you are not in a good postion to berate
someone else for their alleged idiocy. The above remark is itself
strikingly moronic given that James Harris & co post on this newsgroup.

"I find X to be unconvincing" is something you say over, and over, and
over, and over. I find you to be unconvincing. You don't seem to want
to learn mathematics, just to give yourself a reason to adopt superior
airs by pissing all over it.

Why don't you toddle off and learn something?
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 9:35 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Gene Ward Smith wrote:

Quote:

Hatto von Aquitanien wrote:
Gene Ward Smith wrote:


You find mathematics as a whole unconvincing,

Yours is among the most moronic statements I have read in this newsgroup,
and you have some tough competition.

You've made dumb remarks (as well as remarks both rude and stupid, like
the above) all too often, so you are not in a good postion to berate
someone else for their alleged idiocy. The above remark is itself
strikingly moronic given that James Harris & co post on this newsgroup.

You're not very good at the basic logic of sets, I see.

--
Nil conscire sibi
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Sat Jul 08, 2006 9:50 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:

Quote:
You're not very good at the basic logic of sets, I see.

How the hell would you know? I at least took a course in the subject,
which I doubt you ever did.
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 10:17 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Gene Ward Smith wrote:

Quote:

Hatto von Aquitanien wrote:

You're not very good at the basic logic of sets, I see.

How the hell would you know?

Quod erat demonstrandum.

Quote:
I at least took a course in the subject, which I doubt you ever did.

Consider asking for a refund.

--
Nil conscire sibi
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Jürgen R.
science forum beginner


Joined: 06 Feb 2006
Posts: 12

PostPosted: Sat Jul 08, 2006 10:55 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien
<abbot@AugiaDives.hre> wrote:

Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable?

It is one of the favorite playground of mathematics quacks.

Quote:
Quite honestly, I find the
second diagonal method unconvincing.

Have you considered that this mayy be a limitation on your part that
has nothing to do with the validity of the argument?

Quote:
There are a few directions from which
one might attempt to refute his argument. But before I spend a lot to time
trying to formulate my own argument, it seems reasonable to seek prior art.
Can anybody suggest a source which examines this topic?

Yes, but I won't.
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 11:09 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Jürgen Ren wrote:

Quote:
On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien
abbot@AugiaDives.hre> wrote:

I'm interested to know what attempts have been made to refute Cantor's
proof that the real numbers are not denumerable?

It is one of the favorite playground of mathematics quacks.

Such as Frege, Weyl, Witgenstein, Poincarè, Brouwer...?

Quote:
Quite honestly, I find the
second diagonal method unconvincing.

Have you considered that this mayy be a limitation on your part that
has nothing to do with the validity of the argument?

It might be. I find it reassuring, however, that Hermann Weyl appears to
have held a very similar view to mine in this regard. A fact of which I
was unaware when I began this thread.

Quote:
There are a few directions from which
one might attempt to refute his argument. But before I spend a lot to
time trying to formulate my own argument, it seems reasonable to seek
prior art. Can anybody suggest a source which examines this topic?

Yes, but I won't.

As you will.

--
Nil conscire sibi
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David C. Ullrich
science forum Guru


Joined: 28 Apr 2005
Posts: 2250

PostPosted: Sat Jul 08, 2006 11:38 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien
<abbot@AugiaDives.hre> wrote:

Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from which
one might attempt to refute his argument.

Guffaw.

Quote:
But before I spend a lot to time
trying to formulate my own argument, it seems reasonable to seek prior art.
Can anybody suggest a source which examines this topic?

The only such source I'm aware of is sci.math. The refutations you
find here make exactly as much sense as someone "refuting" the
construction of the reals from the rationals via dedekind cuts
by pointing out that pi is not rational. Ie, they exhibit a
basic misunderstanding of the argument that they puport to refute.

************************

David C. Ullrich
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Aatu Koskensilta
science forum Guru Wannabe


Joined: 17 May 2005
Posts: 277

PostPosted: Sat Jul 08, 2006 12:28 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from which
one might attempt to refute his argument. But before I spend a lot to time
trying to formulate my own argument, it seems reasonable to seek prior art.
Can anybody suggest a source which examines this topic?

The topic occasionally crops up in sci.logic and sci.math, as well as
some other newsgroups. Going trough the archives you'll find any number
of inspiring refutations of Cantor's diagonal proof. Proving that ZFC is
inconsistent is another favourite pastime you could try your hand at.

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 1:00 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Aatu Koskensilta wrote:

Quote:
Hatto von Aquitanien wrote:
I'm interested to know what attempts have been made to refute Cantor's
proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from
which
one might attempt to refute his argument. But before I spend a lot to
time trying to formulate my own argument, it seems reasonable to seek
prior art. Can anybody suggest a source which examines this topic?

The topic occasionally crops up in sci.logic and sci.math, as well as
some other newsgroups. Going trough the archives you'll find any number
of inspiring refutations of Cantor's diagonal proof. Proving that ZFC is
inconsistent is another favourite pastime you could try your hand at.

Much to my surprise I have discovered there have been many prominent
thinkers who were not persuaded by Cantor's proposition.

"For if one person can see it as a paradise of mathematicians, why should
not another see it as a joke?" - Ludwig Wittgenstein

--
Nil conscire sibi
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Aatu Koskensilta
science forum Guru Wannabe


Joined: 17 May 2005
Posts: 277

PostPosted: Sat Jul 08, 2006 1:24 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
Much to my surprise I have discovered there have been many prominent
thinkers who were not persuaded by Cantor's proposition.

Sure. But they weren't idiotic enough to think Cantor's proof was
flawed. Rather, they refused to accept the concept of an arbitrary
infinite set and the hierarchy of transfinite numbers.

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 1:38 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Aatu Koskensilta wrote:

Quote:
Hatto von Aquitanien wrote:
Much to my surprise I have discovered there have been many prominent
thinkers who were not persuaded by Cantor's proposition.

Sure. But they weren't idiotic enough to think Cantor's proof was
flawed. Rather, they refused to accept the concept of an arbitrary
infinite set and the hierarchy of transfinite numbers.

quote

url='http://uk.geocities.com/frege%40btinternet.com/cantor/wittgensteinquotes.htm#rfm>

Remarks on the Foundations of Mathematics

V. 7. Imagine set theory's having been invented by a satirist as a kind of
parody on mathematics. ? Later a reasonable meaning was seen in it and it
was incorporated into mathematics. (For if one person can see it as a
paradise of mathematicians, why should not another see it as a joke?) p.
264

(cf Hilbert, D. Uber das Unendliche. Mathematische Annalen 95 (1926 In
Putnam / Benacerraf 183-201, p.191) "No one shall drive us out of the
paradise which Cantror has created for us".

II.15 A clever man got caught in this net of language! So it must be an
interesting net.

II.16 The *mistake* *begins* *when* *one* *says* *that* *the* *cardinal*
*numbers* *can* *be* *ordered* *in* *a* *series* . For what concept has
one of this ordering? One has of course a concept of an infinite series,
but here that gives us at most a vague idea, a guiding light for the
formation of a concept. For the concept itself is abstracted from this and
from other series; or: the expression stands for a certain analogy between
cases, and it can e.g. be used to define provisionally a domain that one
wants to talk about.

That, however, is not to say that the question: "Can the set R be ordered in
a series?" has a clear sense. For this question means e.g.: Can one do
something with these formations, corresponding to the ordering of the
cardinal numbers in a series? Asked: "Can the real numbers be ordered in a
series?" the conscientious answer might be "For the time being I can't form
any precise idea of that". ? "But you can order the roots and the algebraic
numbers for example in a series; so you surely understand the
expression!" ? To put it better, I have got certain analogous formations,
which I call by the common name 'series'. But so far I haven't any certain
bridge from these cases to that of 'all real numbers'. Nor have I any
general method of of trying whether such-and-such a set 'can be ordered in
a series'.

Now I am shewn the diagonal procedure and told: "Now here you have the proof
that this ordering can't be done here". But I can reply "I don't know ? to
repeat ? what it is that can't be done here". Though I can see that you
want to show a difference between the use of "root", "algebraic number",
&c. on the one hand, and "real number" on the other. Such a difference as,
e.g. this: roots are called "real numbers", and so too is the diagonal
number formed from the roots. And similarly for all series of real
numbers. For this reason it makes no sense to talk about a "series of all
real numbers", just because the diagonal number for each series is also
called a "real number". ? Would this not be as if any row of books were
itself ordinarily called a book, and now we said: "It makes no sense to
speak of 'the row of all books', since this row would itself be a book."

II.17. Here it is very useful to imagine the diagonal procedure for the
production of a real number as having been well known before the invention
of set theory, and familiar even to school-children, as indeed might very
well have been the case. For this changes the aspect of Cantor's
discovery. The discovery might very well have consisted merely in the
interpretation of this long familiar elementary calculation.

II.18. For this kind of calculation is itself useful. The question set
would be perhaps to write down a decimal number which is different from the
numbers:

0.1246798 ?
0.3469876 ?
0.0127649 ?
0.3426794 ?
????? (Imagine a long series)

The child thinks to itself: how am I to do this, when I should have to look
at all the numbers at once, to prevent what I write down from being one of
them? Now the method says: Not at all: change the first place of the first
number, the second of the second one &c. &c., and you are sure of having
written down a number that does not coincide with any of the given ones.
The number got in this way might always be called the diagonal number.

II.21 Our *suspicion* ought always to be *aroused* when a *proof* *proves*
*more* *than* *its* *means* *allow* it. Something of this sort might be
called 'a *puffed-up* *proof* '.
</quote>

--
Nil conscire sibi
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