Again, this question is related (\*\*) to a [previous one](https://mathoverflow.net/questions/101700/large-cardinals-without-the-ambient-set-theory?rq=1):


in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: they are special ordinals which are minimal with respect to equinumerosity. 


Ordinals and cardinals live happily within standard set theory.
 Nothing wrong with that, but **what about axiomatizing them *directly*, ie without the underlying set theory**? 


What I mean is: develop a first-order theory of some number system (the intended ordinals), such that they are totally ordered, there is an initial limit ordinal, and satisfy induction with respect to formulas in the language of ordinal arithmetic (here, of course, one will have to adjust the induction schema to accommodate the limit case).


The cardinals could be introduced by adding an order-preserving operator $K$ on the ordinal numbers mimicking their definition in ZFC: cardinals would then be the fixed points of $K$.


1. has some direct axiomatization along these lines been fully developed? I would suspect that the answer is in the affirmative, but I have no refs.
2. the induction schema would be limited to first order formulae, so, assuming that the answer to 1 is yes, is there a theory of non-standard models of Ordinal Arithmetic?
3. Assuming 1 AND 2, what about weaker induction schemas for ordinals (\*)?


(\*) I am thinking again of formal arithmetics and the various sub-systems of Peano


(\*\*) it is not the same, though: here I am asking for a direct axiomatization of ordinals, and indirectly of cardinals, via the operator $k$



Direct axiomatization of ordinal and cardinal numbers
Are these papers of Takeuti the sort of thing you want?


MR0086751 (19,237e) 02.0X
Takeuti, Gaisi,
On the theory of ordinal numbers.
J. Math. Soc. Japan 9 (1957), 93–113.


MR0099918 (20 #6354) 02.00
Takeuti, Gaisi,
On the theory of ordinal numbers. II.
J. Math. Soc. Japan 10 1958 106–120


MR0197302 (33 #5467) 02.18
Takeuti, Gaisi,
A formalization of the theory of ordinal numbers.
J. Symbolic Logic 30 1965 295–317


