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Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

Posted: Sat Jul 08, 2006 8:06 am    Post subject: Attempts to Refute Cantor's Uncountability Proof?

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?
--
Nil conscire sibi
Gene Ward Smith
science forum Guru

Joined: 08 Jul 2005
Posts: 409

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

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.
Patrick1182

Joined: 01 Feb 2006
Posts: 55

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

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?
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

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

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
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

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

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
Gene Ward Smith
science forum Guru

Joined: 08 Jul 2005
Posts: 409

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

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?
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

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

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
Gene Ward Smith
science forum Guru

Joined: 08 Jul 2005
Posts: 409

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

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.
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

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

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.

--
Nil conscire sibi
Jürgen R.
science forum beginner

Joined: 06 Feb 2006
Posts: 12

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

On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien

 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.
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

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

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
David C. Ullrich
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

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

On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien

 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
Aatu Koskensilta
science forum Guru Wannabe

Joined: 17 May 2005
Posts: 277

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

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
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

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

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
Aatu Koskensilta
science forum Guru Wannabe

Joined: 17 May 2005
Posts: 277

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

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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