The Factorial of a Natural Number with Zero

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $n\in\mathbb{N}_{0}$. Multiplication on $\mathbb{N}_{0}$ is associative and commutative with neutral element $1$, by Arithmetic of Multiplication on Omega: Recursion Rules, Distributivity, Associativity, Commutativity, No Zero Divisors, Cancellation and Compatibility with the Order §associative, Arithmetic of Multiplication on Omega: Recursion Rules, Distributivity, Associativity, Commutativity, No Zero Divisors, Cancellation and Compatibility with the Order §commutative and Arithmetic of Multiplication on Omega: Recursion Rules, Distributivity, Associativity, Commutativity, No Zero Divisors, Cancellation and Compatibility with the Order §one.

¶ The factorial of $n$ is the element

$$n!=\prod_{k=1}^{n}k$$

of $\mathbb{N}_{0}$, the product over the interval $\{1,\dots,n\}$ for the multiplication of $\mathbb{N}_{0}$.