Definition 2

Let $n$ be an integer. The product $\prod_{i < n} (i + 1)$ is denoted by $n!$ (read "factorial $n$").

We have $0! = 1$ (Chapter II, § 5, no. 3) and $1! = 1$. For each integer $n$ we have $(n + 1)! = n!\,(n + 1)$. This relation, together with the relation $0! = 1$, characterizes the term $n!$, as is easily seen by induction on $n$.