 Solution

A set $A$ is a countable set if there exists an injective function $A\to \mathbb{N}$, with $\mathbb{N}=\{0,1,2,\dots \}$. Equivalently, it is a finite set or has a bijection with $\mathbb{N}$. An infinite countable set is called a countably infinite set.

The empty set is included: the empty function into $\mathbb{N}$ is injective. This convention distinguishes “countable” from “countably infinite” and will also cover the finite threshold sets in the last part.