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

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Dale Hurliman

Posts: 2
Registered: 12/13/04
Re: Problem with Cantor's diagonal argument
Posted: Feb 14, 2002 9:00 AM
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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
Read Problem with Cantor's diagonal argument
Henry
2/13/02
Read Re: Problem with Cantor's diagonal argument
Andy Berget
2/14/02
Read Re: Problem with Cantor's diagonal argument
Mike Oliver
2/14/02
Read Re: Problem with Cantor's diagonal argument
Doug Norris
2/14/02
Read Re: Problem with Cantor's diagonal argument
Keith Keller
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Mike Oliver
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Bob Kolker
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Seth Dutter
2/14/02
Read Re: Problem with Cantor's diagonal argument
mareg@mimosa.csv.warwick.ac.uk
2/14/02
Read Re: Problem with Cantor's diagonal argument
Nico Benschop
2/14/02
Read Re: Problem with Cantor's diagonal argument
mareg@mimosa.csv.warwick.ac.uk
2/14/02
Read Re: Problem with Cantor's diagonal argument
Willondon
2/14/02
Read Re: Problem with Cantor's diagonal argument
Henry
2/14/02
Read Re: Problem with Cantor's diagonal argument
magidin@math.berkeley.edu
2/15/02
Read Re: Problem with Cantor's diagonal argument
Doug Magnoli
2/14/02
Read Re: Problem with Cantor's diagonal argument
mareg@mimosa.csv.warwick.ac.uk
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Nico Benschop
2/15/02
Read Re: Problem with Cantor's diagonal argument (re finite case)
Nico Benschop
2/15/02
Read cancel <3C6CD566.97EA8F20@chello.nl>
Nico Benschop
2/15/02
Read Re: Problem with Cantor's diagonal argument
Nico Benschop
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Herman Jurjus
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dave Seaman
2/15/02
Read Re: Problem with Cantor's diagonal argument
Jon and Mary Frances Miller
2/15/02
Read Re: Problem with Cantor's diagonal argument
Torkel Franzen
2/15/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/15/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/15/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/15/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/15/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/18/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/18/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/19/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/19/02
Read Re: Problem with Cantor's diagonal argument
Virgil
2/19/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
3/4/02
Read Re: Problem with Cantor's diagonal argument
Alexey Dejneka
3/4/02
Read Re: Problem with Cantor's diagonal argument
Torkel Franzen
3/4/02
Read Re: Problem with Cantor's diagonal argument
Alan Stern
2/16/02
Read Re: Problem with Cantor's diagonal argument
Chip Eastham
2/20/02
Read Re: Problem with Cantor's diagonal argument
SRK
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dale Hurliman
2/14/02
Read Re: Problem with Cantor's diagonal argument
Randy Poe
2/14/02
Read Re: Problem with Cantor's diagonal argument
Henry
2/14/02
Read Re: Problem with Cantor's diagonal argument
Randy Poe
2/14/02
Read Re: Problem with Cantor's diagonal argument
Wade Ramey
2/14/02
Read Re: Problem with Cantor's diagonal argument
nospam@auerbachatunity.ncsu.edu
2/14/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/15/02
Read Re: Problem with Cantor's diagonal argument
Chris Menzel
2/15/02
Read Re: Problem with Cantor's diagonal argument
Dudley Brooks
2/14/02
Read Re: Problem with Cantor's diagonal argument
Phil Carmody
2/14/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/14/02
Read Re: Problem with Cantor's diagonal argument
Jim Heckman
2/15/02
Read Re: Problem with Cantor's diagonal argument
Randy Poe
2/15/02
Read Re: Problem with Cantor's diagonal argument
LarryLard
2/18/02
Read Re: Problem with Cantor's diagonal argument
Harlan Messinger
2/14/02
Read Re: Problem with Cantor's diagonal argument
George Greene
2/15/02
Read Re: Problem with Cantor's diagonal argument
Duran Castore
2/18/02
Read Re: Problem with Cantor's diagonal argument
Jonathan Hoyle

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