116c- Lecture 5

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 \alpha^{\cdot\beta}, we consider the set \{f\in{}^\beta\alpha:f(\xi)=0\mbox{ for all but finitely many }\xi<\beta\} and order it by setting f<g, for f\ne g in this set, iff f(\xi)<g(\xi) as ordinals, where \xi<\beta is the largest ordinal such that f(\xi)\ne g(\xi). In particular, there is such an ordinal \xi. In class, I mentioned that \xi 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 {\sf ORD} 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 {\sf ORD} (for example, one can define the lexicographic ordering on 2\times{\sf ORD}, which would correspond to the “long ordinal” {\sf ORD}+{\sf ORD}). In {\sf ZFC} this is cumbersome (since classes are formulas) but possible. There is an extension of {\sf GB} that allows these operations to be carried out in a more natural way, Morse-Kelley set theory {\sf MK}, 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 {\sf ORD} would also provide a way of extending V 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.”

2 Responses to 116c- Lecture 5

  1. Domenic says:

    Hi Professor,

    Could you explain what was meant by “continuous to the right,” and how this was the usual way to define things? Is it just that the limit “from above” (i.e. counting downward… this is terribly informal, I know) is equal to the number defined? Or is it a commutativity thing?


  2. I think what I meant was simply that the function of two variables S(\alpha,\beta)=\alpha+\beta is continuous on the right (or right-continuous), meaning that for any \beta, \lim_{\gamma\to\beta}S(\alpha,\gamma)=S(\alpha,\beta). (And similarly for multiplication and exponentiation.)

    This is actually the case, although we didn’t formally verify it. The limits here are computed in the topological space {\sf ORD} (with the order topology), which is Hausdorff so limits are well-defined if they exist. One can check that any successor ordinal is isolated (\{\alpha+1\}=(\alpha,\alpha+2)), so the limit expression is only saying something when \beta is a limit ordinal. In this case, since \null [0,\beta]=(-\infty,\beta+1) is an open neighborhood of \beta, \lim_{\gamma\to\beta} means the same that a limit from the left, i.e., with \gamma<\beta, and since S is increasing, then the limit is actually a sup.

    Unfortunately, S is not continuous, because it is not continuous on the left, i.e., S(\alpha,\beta) and \lim_{\gamma\to\alpha}S(\gamma,\beta) need not coincide. For example, n+\omega=\omega for all n\in\omega, while \omega+\omega>\omega.

    Similar remarks hold for multiplication and exponentiation. I believe that multiplication is defined in the somewhat bizarre way it is (so \alpha\cdot\beta=\mbox{ot}(\beta\times\alpha,<_{lex})) so it is continuous on the right rather than on the left, just as addition and exponentiation; notice that multiplication would have been continuous on the left if we had set things to make the more natural identity \alpha\cdot\beta=\mbox{ot}(\alpha\times\beta,<_{lex}) true.

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