Math Insight

Countably infinite definition

A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

For example, the set of integers $\{0, 1, -1, 2,-2, 3, -3, \ldots \}$ is clearly infinite. However, as suggested by the above arrangement, we can count off all the integers. Counting off every integer will take forever. But, if you specify any integer, say $-10,234,872,306$, we will get to this integer in the counting process in a finite amount of time.

Sometimes, we can just use the term “countable” to mean countably infinite. But to stress that we are excluding finite sets, we usually use the term countably infinite.

Countably infinite is in contrast to uncountable, which describes a set that is so large, it cannot be counted even if we kept counting forever.