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Topic: another boring critisism of Cantor's Theorem
Replies: 88   Last Post: May 3, 2004 6:56 AM

 Messages: [ Previous | Next ]
 A N Niel Posts: 2,255 Registered: 12/7/04
Re: another boring critisism of Cantor's Theorem
Posted: Mar 29, 2004 1:04 PM

Feinstein <cafeinst@msn.com> wrote:

> Everyone I am sure is acquainted with Cantor's theorem that the number
> of reals is greater than the number of natural numbers and the
> diagonal argument which proves this. Basically, it says aleph_0 <
> 2^aleph_0, where aleph_0 is the cardinality of the set of natural
> numbers (and 2^aleph_0 is the cardinality of the reals).
>
> I have no problem with his argument, except that I can give another
> equally convincing argument that aleph_0=2^aleph_0: Suppose that
> 2^aleph_0>aleph_0: This implies that aleph_0 = log (2^aleph_0) >log
> (aleph_0), where "log" (base 2) is the inverse of the "2 to the power"
> function generalized for sets of infinite cardinality.

and why should such an inverse exist?

> Then if log
> (aleph_0) is an infinite cardinal number, then aleph_0 is not the
> least infinite cardinal number, which contradicts the definition of
> aleph_0. And if log (aleph_0) is a finite number, then 2^(log
> (aleph_0))=aleph_0 is a finite number, which contradicts its
> definition too. Therefore, aleph_0=2^aleph_0. But Cantor's theorem
> says the opposite of this!

actually, we conclude log(aleph_0) does not exist. That is, there is
no cardinal x such that 2^x = aleph_0.

>
> Therefore, by playing around with infinite numbers, one is playing
> with fire: One is opening up pandora's box to lots of contradictions
>
> Dr. Ben Zona, Ph.D. M.D. Hon.Doc.

Actually, the problem is "playing" with log before defining it and
proving its properties.

Date Subject Author
3/29/04 Craig Feinstein
3/29/04 magidin@math.berkeley.edu
3/29/04 A N Niel
3/29/04 Craig Feinstein
3/29/04 Will Twentyman
3/30/04 Robin Chapman
3/30/04 Jose Carlos Santos
3/30/04 David McAnally
4/3/04 Richard Sabey
3/31/04 Jesse F. Hughes
3/31/04 Lee Rudolph
3/31/04 Jesse F. Hughes
3/31/04 Eckard Blumschein
3/31/04 Jesse F. Hughes
3/31/04 Lee Rudolph
3/31/04 Jesse F. Hughes
3/31/04 Detlef Mueller
4/1/04 Eckard Blumschein
4/1/04 Detlef Mueller
4/1/04 Eckard Blumschein
4/1/04 Herman Jurjus
4/2/04 Eckard Blumschein
4/2/04 Hermann Kremer
4/5/04 Eckard Blumschein
4/6/04 Hermann Kremer
4/7/04 Eckard Blumschein
4/2/04 Hermann Kremer
4/6/04 Eckard Blumschein
4/1/04 ZZBunker
4/1/04 David McAnally
4/1/04 Eckard Blumschein
4/1/04 Robin Chapman
4/1/04 Hermann Kremer
4/2/04 Eckard Blumschein
3/31/04 Robin Chapman
3/31/04 Hermann Kremer
4/1/04 Jesse F. Hughes
4/1/04 Hermann Kremer
4/1/04 Eckard Blumschein
3/31/04 Thomas Nordhaus
4/1/04 Detlef Mueller
4/1/04 Eckard Blumschein
4/1/04 Hermann Kremer
4/2/04 Eckard Blumschein
4/2/04 Jesse F. Hughes
4/1/04 Justin Davis
4/2/04 Eckard Blumschein
4/2/04 Robin Chapman
3/31/04 David McAnally
3/31/04 David Petry
3/31/04 Torkel Franzen
4/2/04 David Petry
4/2/04 Torkel Franzen
4/1/04 Rupert
4/2/04 David Petry
4/2/04 David C. Ullrich
4/3/04 Torkel Franzen
4/3/04 David C. Ullrich
4/4/04 David Petry
4/4/04 Daryl McCullough
4/5/04 Rupert
4/5/04 David Petry
4/6/04 Rupert
4/6/04 Rupert
4/7/04 David Petry
4/7/04 Rupert
4/9/04 Herman Jurjus
4/9/04 Rupert
4/13/04 David Petry
4/13/04 Rupert
4/14/04 David Petry
4/14/04 Rupert
4/14/04 Rupert
4/16/04 Herman Jurjus
4/16/04 Rupert
4/22/04 Eckard Blumschein
5/3/04 Herman Jurjus
4/6/04 Herman Rubin
4/6/04 David Petry
4/7/04 Rupert
4/7/04 Rupert
4/5/04 David C. Ullrich
4/6/04 Herman Rubin
4/3/04 Rupert
3/31/04 Ted Hwa
3/29/04 David Manheim
3/29/04 Jesse F. Hughes
3/29/04 Lee Rudolph
3/29/04 Jesse F. Hughes