and recursions of countable length, the weaker axiom of dependent choice is sufficient. Because there are models of Zermelo–Fraenkel set theory of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful. See also Mathematical induction ∈-induction Transfinite number Well-founded induction Zorn's lemma Notes References External links Mathematical induction Ordinal numbers Recursion