I'm considering the natural numbers to be the nonnegative integers. The principle of strong induction can be stated as follows,
"If $P$ is a property such that for any $x$, if $P$ holds for all natural numbers less then $x$, then $P$ holds for $x$ as well, then $P$ holds for all natural numbers $x$."
This doesn't require a base case. My question is why? Shouldn't $P$ hold for $0$ for the induction to work?