## Most recent change of CountableSet

Edit made on August 13, 2011 by ColinWright at 12:07:14

Deleted text in red / Inserted text in green

WW WM
HEADERS_END
Since counting is the act of assigning a natural number to each item,
Georg Cantor defined a set as countably infinite if it can be put in
one-one correspondence with the natural numbers.

Using this definition, trivially the natural numbers themselves are
countably infinite, but so are the even numbers, the integers and
the square numbers.

!! What about the Rational Numbers?

Form an (infinite) grid like this. Only the first six rows and columns are shown.

| 1/1 | 1/2 | 1/3 | 1/4 | 1/5 | 1/6 | ... |
| 2/1 | 2/2 | 2/3 | 2/4 | 2/5 | 2/6 |
| 3/1 | 3/2 | 3/3 | 3/4 | 3/5 | 3/6 |
| 4/1 | 4/2 | 4/3 | 4/4 | 4/5 | 4/6 |
| 5/1 | 5/2 | 5/3 | 5/4 | 5/5 | 5/6 |
| 6/1 | 6/2 | 6/3 | 6/4 | 6/5 | 6/6 |
| /etc/ |

We want to show that these can be put into one-to-one correspondence with the natural numbers,
and there are two easy ways to think about this.

COLUMN_START^
Zig-zag back and forth through the above table:

* 1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, 3/2, 4/1, 5/1, 4/2, 3/3, 2/4, 1/5, 1/6, 2/5, ...
** !/ Put your finger on the table and trace the path that these makes ... !/

COLUMN_SPLIT^
List all those whose numerator and denominator add to two, then to three, then to four, etc, like this:

* 1/1,
* 1/2, 2/1,
* 1/3, 2,2, 2/2, 3/1,
* 1/4, 2/3, 3/2, 4/1,
* /etc/

COLUMN_END

Each of these two ways will form a list of all the rational numbers. Therefore there exists a one-to-one correspondence with the natural numbers. So the rational numbers are countably infinite.

The size of countable sets is given the transfinite number EQN:\aleph_0 (pronounced "aleph null") (see transfinite numbers)

Are all sets countable? (see Uncountable sets)