Definitions of ordinal besides von Neumann & Frege-Russel?

Question

So my Google-fu didn't show any references on this. I'm studying an obscure set theory (ML, a variation on NF with proper classes) and it seems to not deal well with the standard definitions of ordinal number. I'm wondering if anyone knows of any references (if they exist) to approaches besides the von Neumann or Frege-Russel ones.

(The specific problems are that the von Neumann ordinals are unstratified, so even ones that exist can't seem to measure wellorders [the usual recursion fails]. The Frege-Russel version works, but the numbers can't generally be guaranteed to be sets since quantifiers in set abstraction must be restricted to sets, and some sets have subclasses which are proper classes [so the usual meaning of "wellorder", among other things, would need to be altered].)

Answer 1

You might reconsider whether you actually need ordinals at all. Well-orderings, yes, but perhaps not ordinals, which are at bottom merely a convenience, a way of referring to the isomorphism classes of well-orderings.

The usual von Neumann ordinals, for example, provide in ZF a canonical choice of representative from each isomorphism class of well-orderings. This is very convenient, to be sure, but one may undertake almost all of set theory without those representatives, using just arbitrary well-orders instead. One needs to develop the theory of transfinite recursion in the setting of arbitrary well-orders, rather than ordinals, but everything works fine. For example, this is how Moschovakis proceeds in his introductory set theory book, Notes on Set Theory, which I used once while teaching undergraduate set theory at Berkeley. (See my review of it.)

One counterpoint to this perspective, however, is that the ordinals are convenient — a point I make in my review — and they help to uniformize many transfinite recursive processes. Gödel remarked, for example, that the von Neumann ordinal concept is what allowed him finally to see his way through the thicket when defining the constructible universe: having a single canonical way to choose representatives from the isomorphism classes made the transfinite recursion a bit clearer. With hindsight, however, it seems clear enough that one can do it without those canonical choices, by working with arbitrary well-orders as Moshchovakis does.

Answer 2

If you really are studying Quine's ML you could read the book by Quine whose title gave the name `mathematical logic' to the system you are looking at. IN ML ordinals are isomorphism classes of wellorderings, as they are in NF and in Principia mathematica. Ordinals in this style are discussed throughout the NF literature. (Look at the New Foundations Home page maintained by Randall Holmes); make me a rich man by buying a copy of Set theory with a universal set!

There are also Scott's trick ordinals, but they are no use in ML. Another thing one can do is define ordinals as isomorphism classes locally (living inside some set). Then one has to prove that all the local systems of ordinals that one obtains cohere.

How on earth did you get into ML?