TITLE: Direct axiomatization of ordinal and cardinal numbers
QUESTION [8 upvotes]: Again, this question is  related  (**) to a previous one:
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 $K$.

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. 
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? 
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 $k$

REPLY [3 votes]: You might consider taking a look at a paper by Athanassios Tzouvaras titled "Cardinality without enumeration" (look under title on the Web), especially at Definition 3.1.  Since it is short, I will quote it verbatim:
"Definition 3.1  Let $M$ be a model of $ZF$.  A notion of cardinality for $M$ is a mapping $C$$\subset$$M$ such that:
(1) dom($C$)=$M$ and  r ng($C$)=Card
(2) $C$($\kappa$)=$\kappa$ for every $\kappa$$\in$_Card_
(3) For any disjoint sets $x$, $y$ $C$($x$$\cup$$y$)=$C$($x$)$+$$C$($y$)
(4) For any $x$, $y$ $C$($x$$\times$$y$)=$C$($x$) $\cdot$ $C$($y$)
(5) If $f$ : $x$$\rightarrow$$y$ is an injective mapping, then $C$($x$)$\le$$C$($y$)
A cardinality notion $C$ is said to be standard if in addition the converse of (5) holds, i.e. if $C$($x$)$\le$$C$($y$) implies that there is an injective $f$  from $x$ to $y$."
An especially interesting consequence of the cardinal notion 
$C$ being standard is Remark 3.2(ii):
"If $C$ is standard, then $C$($x$)=$C$($y$) implies that there is an injective $f$ from $x$ onto $y$.  So the existence of a standard notion of cardinality implies $AC$ [and the Kunen inconsistency as well--my comment].  Moreover in the presence of $AC$ there is a unique notion of cardinality, the standard one.  Thus in order to depart from the standard notion of cardinality, we must drop $AC$." 
Is Definition 3.1 more along the lines of what you are looking for, at least regarding cardinality?