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](https://en.wikipedia.org/wiki/S_(set_theory)) has a separate variable sort for "stages" with appropriate axioms, but not enough to build a full theory of ordinals.



Ordinals separate from set theory
I think the following article by Takeuti does what you want:


Takeuti, Gaisi, A formalization of the theory of ordinal numbers. J. Symbolic Logic 30 1965 295–317.


The Mathematical Reviews / MathSciNet label for it is MR0197302 (33 #5467).


