Direct axiomatization of ordinal and cardinal numbers

Question

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 $K$.

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 $k$

Answer 1

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

Answer 2

This is not exactly what you asked for, but there is an interesting axiomatization of Ordinals + Sets of Ordinals, which turns out to be precisely equiconsistent with ZFC.

• Peter Koepke, Martin Koerwien, The Theory of Sets of Ordinals.

Basically, if one is committed to the ordinals and having certain kinds of sets of ordinals, then you can build Gödel's constructible universe $L$ and simulate a model of ZFC that way.

Answer 3

I would like to suggest the theory of "ordinal algebras" and "cardinal algebras". There are books with the same title by Tarski.

Mathscinet review of the book cardinal algebras:

This book is an axiomatic investigation of the novel types of algebraic systems which arise from three sources: the arithmetic of cardinal numbers; the formal properties of the direct product decompositions of algebraic systems; the algebraic aspects of invariant measures, regarded as functions on a field of sets.

Mathscinet review of the book ordinal algebras:

An ordinal algebra is an additively written, but not ordinarily commutative, associative system, equipped with a suitably axiomatized operation of simply infinite addition, $∑^∞_1a_ν$, and an operation of conversion, $a^∗$. (Infinite addition is characterized and employed "only insofar as [it is] involved in the study of finite addition''.) The additive theory of order types provides a familiar application, although far from exhausting the interest of the theory. The theory of ordinal algebras differs from that presented in the author's "Cardinal algebras'' principally in the non-commutativity of addition.

Answer 4

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?