We defined addition, multiplication, and exponentiation of ordinals, and stated some basic properties of these operations. They extend into the transfinite the usual operations on natural numbers.
I made a mistake when indicating how to define these operations “intrinsically” rather than as a consequence of the transfinite recursion theorem: In the definition of ordinal exponentiation , we consider the set and order it by setting , for in this set, iff as ordinals, where is the largest ordinal such that . In particular, there is such an ordinal . In class, I mentioned that was the smallest such ordinal, but this does not work.
Using Hartog’s function and transfinite recursion we defined the long sequence of (well-ordered) cardinals, the alephs.
Remark. It was asked in class whether one can make sense of well-orders longer than and if one can extend to them the operations we defined.
Of course, one can define classes that are well-orders of order type longer than (for example, one can define the lexicographic ordering on , which would correspond to the “long ordinal” ). In this is cumbersome (since classes are formulas) but possible. There is an extension of that allows these operations to be carried out in a more natural way, Morse-Kelley set theory , briefly discussed here.
However, I do not know of any significant advantages of this approach. But a few general (and unfortunately vague) observations can be made:
- Most likely, any way of extending well-orders beyond would also provide a way of extending to a longer “universe of classes.” The study of these end-extensions (in the context of large cardinals, where it is easier to formalize these ideas) has resulted in an interesting research area originated by Keisler and Silver with recent results by Villaveces and others.
- I also expect that any systematic way of doing this would translate with minor adjustments into a treatment of indiscernibles and elementary embeddings (which could potentially turn into a motivation for the study of these important topics and would be interesting at least from a pedagogical point of view).
- As I said, however, I do not know of any systematic attempt at doing something with these “long ordinals.” With one exception: the work of Reinhardt, with the caveat that I couldn’t make much sense of it in any productive way years ago. But this is an excuse to recommend a couple of excellent papers by Penelope Maddy, Believing the Axioms I and II, that originally appeared in The Journal of Symbolic Logic in 1988 and can be accessed through JSTOR. These papers discuss the intuitions behind the axioms of set theory and end up discussing more recent developments (like large cardinal axioms and determinacy assumptions), and I believe you will appreciate them. During her discussion of very large cardinals, Maddy mentions Reinhardt ideas, so this can also be a place to start if one is interested in the issue of “long ordinals.”