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

 Messages: [ Previous | Next ]
 Henry Posts: 5 Registered: 12/13/04
Problem with Cantor's diagonal argument
Posted: Feb 13, 2002 11:29 PM

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? A possible list would be something
like:

1 <--> .25
2 <--> .50

Here the number 1 is assigned to the specific number .25, not to an
arbitrary number that has an infinite number of digits after the
decimal point.

Given these questions, how can the proof itself hold true. I don't
see that its initial premise makes logical sense. What's also
confusing is saying that if you can't use the integers to count the
decimals that kind of assumes that you'll run out of integers (you'll
reach the largest integer ever) which makes no sense. Comments

In an alternate but related topic, I think it is absurd to talk
about an infinite being larger than another. That just does not make
sense at all. Infinity doesn't have an end so:

Inf + Inf = Inf
Inf * Inf = Inf and so on.

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