Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Problem with Cantor's diagonal argument
Replies: 65   Last Post: Mar 4, 2002 1:36 PM

 Messages: [ Previous | Next ]
 Dale Hurliman Posts: 2 Registered: 12/13/04
Re: Problem with Cantor's diagonal argument
Posted: Feb 14, 2002 9:00 AM

Henry wrote:

> While reading Paul Erdos' story in "My brain is open" I found Cantor's
> diagonal argument which he used to 'prove' that the decimals between 0
> and 1 are uncountable. What's even worse, he used that to say that the
> infinity of the decimals is larger than the infinity of the integers
> (but that's another matter). What I would like to know is how could he
> establish the diagonal argument on a list that I'm not sure can be
> created at all. How can someone create a list like:
>
> 1 <--> .2332245.....
> 2 <--> .4898495.....
>
> The basic question here is how can you assign the number 1 to a
> number that has an infinite number of digits. In the example above,
> what is the number that 1 'points' to? Since you can keep adding an
> infinite number of digits after the decimal point how can you say that
> 1 is pointing to a specific number?
>

<snip>

Cantor's arguement is a kind of _reductio ad absurdum_ arguement. He
says, in essence, "Let us assume one could make a one to one list of the
integers and the corresponding real numbers between 0 and 1. If that
could be done, even then, I can still show that there are more reals in
the range 0-1 than there are integers." And you must grant that that
initial assumption is certainly a liberal assumption. So even granted
such a liberal assumtion there are still more reals than integers.
Therefore our initial liberal assumption must be false. Such a list can
not be made. Therefore, based on the definition of set size (The size
of a set is defined by a one to one correspondence between integers and
set members) there must be more real numbers between 0 and 1 than there
are integers. Q.E.D.
Dale

Date Subject Author
2/13/02 Henry
2/13/02 Andy Berget
2/14/02 Mike Oliver
2/14/02 Doug Norris
2/14/02 Keith Keller
2/14/02 Dudley Brooks
2/14/02 Mike Oliver
2/14/02 Dudley Brooks
2/14/02 Dave Seaman
2/14/02 Dudley Brooks
2/14/02 Dave Seaman
2/14/02 Dudley Brooks
2/14/02 Bob Kolker
2/14/02 Dave Seaman
2/14/02 Seth Dutter
2/14/02 mareg@mimosa.csv.warwick.ac.uk
2/14/02 Nico Benschop
2/14/02 mareg@mimosa.csv.warwick.ac.uk
2/14/02 Willondon
2/14/02 Henry
2/14/02 magidin@math.berkeley.edu
2/15/02 Doug Magnoli
2/14/02 mareg@mimosa.csv.warwick.ac.uk
2/14/02 Dudley Brooks
2/14/02 Nico Benschop
2/15/02 Nico Benschop
2/15/02 Nico Benschop
2/15/02 Nico Benschop
2/14/02 Dave Seaman
2/14/02 Herman Jurjus
2/14/02 Dave Seaman
2/15/02 Jon and Mary Frances Miller
2/15/02 Torkel Franzen
2/15/02 Virgil
2/15/02 Harlan Messinger
2/15/02 Virgil
2/15/02 Harlan Messinger
2/15/02 Virgil
2/18/02 Harlan Messinger
2/18/02 Virgil
2/19/02 Harlan Messinger
2/19/02 Virgil
2/19/02 Dudley Brooks
3/4/02 Alexey Dejneka
3/4/02 Torkel Franzen
3/4/02 Alan Stern
2/16/02 Chip Eastham
2/20/02 SRK
2/14/02 Dale Hurliman
2/14/02 Randy Poe
2/14/02 Henry
2/14/02 Randy Poe
2/14/02 nospam@auerbachatunity.ncsu.edu
2/14/02 Dudley Brooks
2/15/02 Chris Menzel
2/15/02 Dudley Brooks
2/14/02 Phil Carmody
2/14/02 Harlan Messinger
2/14/02 Jim Heckman
2/15/02 Randy Poe
2/15/02 LarryLard
2/18/02 Harlan Messinger
2/14/02 George Greene
2/15/02 Duran Castore
2/18/02 Jonathan Hoyle