Countable infinity

A countably infinite set is any set that can be put into a one-to-one correspondence with the natural numbers.

For example, the set of integers {...,-3,-2,-1,0,1,2,3,...} is countably infinite, since we can pair up every integer with exactly one natural number, as follows:

(0,0), (1,1), (-1,2), (2,3), (-2,4), (3,5), (-3,6), ...

Similarly, the rational numbers are countably infinite, since we can pair up each rational number with a single natural number.

It can be shown that this process is impossible with real numbers. Real numbers are therefore said to be uncountably infinite.