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.

(As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyond projective). To illustrate this, let me "randomly" highlight two examples: See here for $\Sigma^1_2$-Woodin cardinals and, more generally, the noti […]

I am not sure which statement you heard as the "Ultimate $L$ axiom," but I will assume it is the following version: There is a proper class of Woodin cardinals, and for all sentences $\varphi$ that hold in $V$, there is a universally Baire set $A\subseteq{\mathbb R}$ such that, letting $\theta=\Theta^{L(A,{\mathbb R})}$, we have that $HOD^{L(A,{\ma […]

A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\in\Gamma$. Note that if $\Gamma\subseteq\mathcal P(\mathbb R)$ and $L(\Gamma,\mathbb R)\models \Gamma=\mathcal P(\mathbb R)$, then $\Gamma$ is a Wadge initial se […]

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Yes. Consider, for instance, Conway's base 13 function $c$, or any function that is everywhere discontinuous and has range $\mathbb R$ in every interval. Pick continuous bijections $f_n:\mathbb R\to(-1/n,1/n)$ for $n\in\mathbb N^+$. Pick a strictly decreasing sequence $(x_n)_{n\ge1}$ converging to $0$. Define $f$ by setting $f(x)=0$ if $x=0$ or $\pm x_n […]

All proofs of the Bernstein-Cantor-Schroeder theorem that I know either directly or with very little work produce an explicit bijection from any given pair of injections. There is an obvious injection from $[0,1]$ to $C[0,1]$ mapping each $t$ to the function constantly equal to $t$, so the question reduces to finding an explicit injection from $C[0,1]$ to $[ […]

One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]

"There are" examples of discontinuous homomorphisms between Banach algebras. However, the quotes are there because the question is independent of the usual axioms of set theory. I quote from the introduction to W. Hugh Woodin, "A discontinuous homomorphism from $C(X)$ without CH", J. London Math. Soc. (2) 48 (1993), no. 2, 299-315, MR1231 […]

This is Hausdorff's formula. Recall that $\tau^\lambda$ is the cardinality of the set ${}^\lambda\tau$ of functions $f\!:\lambda\to\tau$, and that $\kappa^+$ is regular for all $\kappa$. Now, there are two possibilities: If $\alpha\ge\tau$, then $2^\alpha\le\tau^\alpha\le(2^\alpha)^\alpha=2^\alpha$, so $\tau^\alpha=2^\alpha$. In particular, if $\alpha\g […]

Fix a model $M$ of a theory for which it makes sense to talk about $\omega$ ($M$ does not need to be a model of set theory, it could even be simply an ordered set with a minimum in which every element has an immediate successor and every element other than the minimum has an immediate predecessor; in this case we could identify $\omega^M$ with $M$ itself). W […]

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.