What Is Countable Infinity?

Countable infinity refers to an infinite set whose elements can be arranged in a sequence, like the natural numbers. A set is called countably infinite if there exists a one-to-one correspondence between it and the set of natural numbers.

Not only the set of natural numbers, but also the sets of integers and rational numbers are countably infinite, since their elements can be listed in a sequence through a one-to-one mapping. Although these sets contain infinitely many elements, they are still considered "countable" because it is possible to assign a unique number to each element.

The size of such countably infinite sets is denoted by the symbol $\aleph_0$ (read as "aleph-null" or "aleph-zero"). This is the first cardinal number used to describe the size of infinite sets in set theory.

In contrast, the set of real numbers cannot be put into a one-to-one correspondence with the natural numbers. Real numbers cannot be fully listed in a sequence, and for that reason, they are classified as uncountably infinite. This fact can be demonstrated using Cantor’s diagonal argument.

Infinity comes in different sizes, and countable infinity represents the smallest level of infinity.