TITLE: Definitions of ordinal besides von Neumann & Frege-Russel?
QUESTION [8 upvotes]: 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].)

REPLY [8 votes]: 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?