Explain carefully what is meant by a well-founded relation on a set. State the recursion theorem, and use it to prove that a binary relation $r$ on a set $a$ is well-founded if and only if there exists a function $f$ from $a$ to some ordinal $\alpha$ such that $(x, y) \in r$ implies $f(x)<f(y)$.

Deduce, using the axiom of choice, that any well-founded relation on a set may be extended to a well-ordering.