I am reading "Set theory, logic and their limitations" by Moshe Machover (page 264).
If $\mathfrak {^*N}$ is an $\mathcal L$-structure and $X$ is any subset of $^*N$, we say that $X$ is inductive in $\mathfrak {^*N}$ if it satisfies the condition:
If $^*0 \in X$, and for every $x \in X$ also $^*s(x) \in X$, then $X =$ $^*N$
It goes on to say that the Principle of induction states that every subset of $N$ is inductive (in the standard interpretation).
However, this definition seems a bit different from some other definitions I've seen that seem to remove the "if" qualifier?