Remark

Let E be a set, let A be a subset of E, let $g$ be a mapping of A into E, and let $a$ be an element of A. Take $S\{u\}$ to be the term $g(u)$. Criterion C63 is applicable and proves the existence of a mapping $f$ of $\mathbf{N}$ onto a set V such that $f(0) = a$ and $f(n + 1) = g(f(n))$ for every integer $n$. It may happen that $\mathrm{V} \subset \mathrm{A}$; if not, let $p$ be the largest integer such that $f([0, p]) \subset \mathrm{A}$. Then $f(p + 1) = g(p) \notin \mathrm{A}$, and $g(g(p))$ is a term about which nothing can be said. Hence in this case $f$ is considered to be defined only on the interval $[0, p + 1]$ ("restricted induction").