Definition of factorial function for sets

Question

Why is the factorial function expressed in terms of $(n+1)$ for sets?

$0! = 1$
$(n+1)! = (n+1) \times n! $ for all $n$ $\in\mathbb{N}$

Instead of the more "common"

$0! = 1$
$n! = n \times (n-1)!$

Answer 1

Either definition is acceptable. The only difference here is that the first definition assumes that: $$ n \in \mathbb N = \{0, 1, 2, 3, \ldots\} $$ while the second definition assumes that: $$ n \in \mathbb Z^+ = \{1, 2, 3, \ldots\} $$