Countable

A set X is countable if and only if there is a bijection from X to the set of natural numbers. Countable sets include the set of integers and the set of rational numbers. Countable sets are said to have cardinality <math>\aleph</math>.

There is no smallest countable set. Indeed, the set of natural numbers is in bijection with the natural numbers without 0: subtracting 1 from every number gives a bijection from the first set to the second. Repeating this process shows that for any initial segment of the the natural numbers (such as {1, 2, ..., n}), we have a bijection between the set of natural numbers and the set of natural numbers without this segment. The bijection is simply subraction by n.