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 correpsond 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 sytematic 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.”

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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?

I think what I meant was simply that the function of two variables is continuous on the right (or right-continuous), meaning that for any , . (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 (with the order topology), which is Hausdorff so limits are well-defined if they exist. One can check that any successor ordinal is isolated (), so the limit expression is only saying something when is a limit ordinal. In this case, since is an open neighborhood of , means the same that a limit from the left, i.e., with , and since is increasing, then the limit is actually a sup.

Unfortunately, is not continuous, because it is not continuous on the left, i.e., and need not coincide. For example, for all , while .

Similar remarks hold for multiplication and exponentiation. I believe that multiplication is defined in the somewhat bizarre way it is (so ) 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 true.

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Let $s$ be the supremum of the $\mu$-measures of members of $\mathcal G$. By definition of supremum, for each $n$, there is $G_n\in\mathcal G$ with $\mu(G_n)>s-1/n$. Letting $G=\bigcup_n G_n$, then $G\in \mathcal G$ since $\mathcal G$ is closed under countable unions, and $\mu(G)=s$, since it is at least $\sup_n\mu(G_n)$ but it is at most $s$ (by definiti […]

The result you are trying to prove is false. For example, if $a=\omega+1$ and $b=\omega+\omega$, then $a+b=\omega\cdot 3>b$. Here is what is true: first, the key result you should establish (by induction) is that An ordinal $\alpha>0$ has the property that for all $\beta

Very briefly: Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel): MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, […]

This is a very interesting question and the subject of current research in set theory. There are, however, some caveats. Say that a set of reals is $\aleph_1$-dense if and only if it meets each interval in exactly $\aleph_1$-many points. It is easy to see that such sets exist, have size $\aleph_1$, and in fact, if $A$ is $\aleph_1$-dense, then between any tw […]

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?

Thanks!

I think what I meant was simply that the function of two variables is continuous on the right (or

right-continuous), meaning that for any , . (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 (with the order topology), which is Hausdorff so limits are well-defined if they exist. One can check that any successor ordinal is isolated (), so the limit expression is only saying something when is a limit ordinal. In this case, since is an open neighborhood of , means the same that a limit

from the left, i.e., with , and since is increasing, then the limit is actually a sup.Unfortunately, is not continuous, because it is not continuous on the left, i.e., and need not coincide. For example, for all , while .

Similar remarks hold for multiplication and exponentiation. I believe that multiplication is defined in the somewhat bizarre way it is (so ) 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 true.