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A set $S$ is said to be countable if there is an injection $f:S\to\mathbb{Z}$. Informally, a set is countable if it has at most as many elements as does the set of integers. The countable sets can be divided between those which are finite and those which are countably infinite.

The name "countable" arises because the countably infinite sets are exactly those which can be put into bijection with the natural numbers, i.e. those whose elements can be "counted."

Countably infinite sets include the integers, the positive integers and the rational numbers.

Uncountable sets include the real numbers and the complex numbers.


  • Any finite set is countable.