Is there an exposition and development of ordinals theory separate from set theory? That is, some first-order theory where terms are interpreted as ordinals, with constant $0$ (and maybe $\omega$), axioms for taking successor and limit (this might be easier to express in a second-order theory), an axiom schema for transfinite induction and with usual ordinal operations (sum, product, power) defined via transfinite recursion?

*E.g.*, G. Boolos's system S has a separate variable sort for "stages" with appropriate axioms, but not enough to build a full theory of ordinals.