Countably_infinite

Reference mathematics

Something is countably infinite if a one to one mapping

can be made between it and the natural numbers or a subset of the natural numbers.

For example, positive even numbers are countably infinite

as displayed by the mapping:

*2 maps to 1

*4 maps to 2

*6 maps to 3

*...

*N maps to (N/2) where N is a positive even number

Sometimes it is necessary to use more than one mapping.

This is where you map the set you want to show as countably infinite to another set. You then map this other set to the natural numbers.

There are other, uncountable infinites. It is easy to show, for instance, that the set of real numbers is not countably infinite, using Cantor's diagonal slash argument. The countable infinite is the smallest one.