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 $k$

  • $\begingroup$ Do you know Algebraic Set Theory by Joyal and Moerdijk? They give a set-up where the poset of ordinals is given as a suitable initial algebra of an endofunctor. (It might not be exactly what you want, but worth a look nonetheless.) $\endgroup$
    – Todd Trimble
    Feb 7, 2016 at 21:25
  • $\begingroup$ Thanks Todd. No, I know of its existence, but I have never actually studied it. That seems somewhat connected to my question, but ideally I would like Ordinal Arithmetic to be totally independent from a set theory background, even in an algebraic presentation $\endgroup$ Feb 7, 2016 at 21:28

4 Answers 4


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


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.

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.

  • $\begingroup$ Hello Joel, long time no talk! Yes, that is not exactly what I am looking for, but it is nevertheless very (!) interesting to me. As you surely recall, I asked a few years back on the "Ackermann Yoga", ie doing set theory WITHIN arithmetic. Now, I suspect your ref moves along similar lines: IF we have the ordinals, THEN we can simulate ZFC within (we do not need to assume actual real sets, I suspect the Ackermann yoga can get us there. $\endgroup$ Feb 7, 2016 at 22:09
  • $\begingroup$ but then things become even more interesting: suppose you axiomatize Ord in a "weak" induction way (just like you do with sub-systems of Peano), then the encoded set theory would be likely weaker (and odder) than ZFC. Perhaps more set universes ready to be adopted by your fantastic multi-verse... $\endgroup$ Feb 7, 2016 at 22:12
  • 1
    $\begingroup$ Mirco, it's always nice to chat with you. The reason I view this work as one-step removed from your question is that it isn't just a theory of the ordinals, but a theory of sets of ordinals, which makes it already a little set-theoretic. But only one-step removed, and not two, since the theory does not have sets-of-sets of ordinals. $\endgroup$ Feb 7, 2016 at 22:28
  • $\begingroup$ I wonder what you need L for. Sets of ordinals encode up to isomorphism arbitrary relations on (well-orderable) sets, and every set can be recovered from the $\in$ relation on its transitive closure (this is a standard argument), so the theory should be able to completely describe the ZFC universe. What am I missing? $\endgroup$ Feb 9, 2016 at 8:09
  • 1
    $\begingroup$ OK, now I’ve seen the paper. Indeed, they actually show that ZFC is bi-interpretable with SO, as I would expect. $\endgroup$ Feb 9, 2016 at 14:46

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.

  • 2
    $\begingroup$ I have Cardinal Algebras in hard copy. Not that it helps the OP being in a whole other continent. I just really like to tell how I bought it for 2 USD in a pretty reasonable condition from the students library when I was an undergrad. (I didn't even know what's inside, just that it was by Tarski and was about cardinals.) $\endgroup$
    – Asaf Karagila
    Feb 8, 2016 at 23:11
  • $\begingroup$ Unlike you, I looked for it at the library of many universities, but could not find it. $\endgroup$ Feb 14, 2016 at 11:08
  • $\begingroup$ I bet it's not easy to find. I'm sure that you looked at the KGRC's uber-libarary of logic and set theory, so the fact you didn't find it is quite impressive and says a lot about the availability of this book in modern times. $\endgroup$
    – Asaf Karagila
    Feb 14, 2016 at 13:43

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?

  • $\begingroup$ @Mirco A. Mannucci: I wish to add, as a further reference, O. Deiser's paper "An axiomatic theory of well-orderings" (Rev. of Symbolic Logic 4, no. 2, 186-204 (2011)) as (at least) something close to "a direct theory of ordinal numbers" which, as Prof. Geschke states in his answer to Robin Saunders' mathoverflow question " 'Set Theory' founded on lists rather than sets" , is founded on the notion of 'list'. It should be also noted that since Prof. Deiser proved his theory does not go beyond $ZFC$ in (theorem-proving) strength, it should have an adequate theory of cardinal arithmetic as $\endgroup$ Feb 15, 2016 at 11:14
  • $\begingroup$ (cont.) well... $\endgroup$ Feb 15, 2016 at 11:19

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