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.

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 ...

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, 3/1,
  • 1/4, 2/3, 3/2, 4/1,
  • etc

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 $\aleph_0$ (pronounced "aleph null") (see transfinite numbers)

Are all sets countable? (see Uncountable sets)

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