Drexel dragonThe Math ForumDonate to the Math Forum

Search All of the Math Forum:

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

Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Cantor's diagonal argument.
Replies: 24   Last Post: Oct 12, 2001 5:16 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Giles Redgrave

Posts: 3
Registered: 12/13/04
Re: Cantor's diagonal argument.
Posted: Oct 4, 2001 6:39 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Randy Poe <rpoe@nospamatl.lmco.com> wrote in message news:<3BBB4A4D.7025D757@nospamatl.lmco.com>...
> briggs@encompasserve.org wrote:
> > Classic issue. The resolution is to realize that this number is
> > not a natural number. All natural numbers have finitely many
> > decimal digits. This number has infinitely many.

> But it's important to note, as it was in this thread, that
> this property (finite number of digits) should not be
> taken as an axiom. Instead it follows by induction from
> the construction of the natural numbers:
> 1 has a single digit (in all bases).
> Let a be any natural number with a finite
> number of digits, call it n_a. Then (a+1) has
> at most (n_a+1) digits, which is finite.

But can't you use the same argument to show that the number of natural
numbers is finite.

Call the set of integers from 1 to n A_n.

The number of members of A_1 is finite.
If the number of members of A_n is finite then so is the number of
members of A_{n+1} (A_n + 1).
Therefore there are a finite number of natural numbers.

I can't see how the infinity of natural numbers can be represented by
a finite number of digits. In decimal the number of integers
representable by n digits is 10^n. How can 10^n be infinite when n is

Or to put it onother way, doesn't log n exceeds any finite value as n
-> infinity.

I must not understand what is meant by infinite and finite, is there a
clear mathematical definition of these terms. Help, I'm still


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.