User blog comment:LittlePeng9/Higher order set theory/@comment-24509095-20141016112825/@comment-2033667-20141016202501

As discussed on chat, if we take the Von Neumann definition of ordinals, we can let O equal the class On. With this definition, ordinal arithmetic (successors, addition, multiplication, etc.) still works out of the box with no modifications -- so conveniently, we already have solid definitions for O + 1, O + 2, O + O, O^O, epsilon_O, etc. I can't emphasize enough that these are not ordinals, but...uh..."oodinals." Yeah. That's what I'll call them.

I therefore define the following: an oodle X is an oodinal iff it is strictly well-ordered by ∈ and every element of X is a suboodle of X. (A relation is a well-ordering over an oodle iff it is transitive, asymmetric, and total and every nonempty suboodle of the oodle has a minimum according to the relation.)