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 technique of almost disjoint forcing was introduced in MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some applications of almost disjoint sets. In Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pp. 84–104, North-Holland, Amsterdam, 1970. Fix an almost disjoint family $X=(x_\alpha:\alpha

At the moment most of those decisions come from me, at least for computer science papers (those with a 68 class as primary). The practice of having proceedings and final versions of papers is not exclusive to computer science, but this is where it is most common. I've found more often than not that the journal version is significantly different from the […]

The answer is no in general. For instance, by what is essentially an argument of Sierpiński, if $(X,\Sigma,\nu)$ is a $\sigma$-finite continuous measure space, then no non-null subset of $X$ admits a $\nu\times\nu$-measurable well-ordering. The proof is almost verbatim the one here. It is consistent (assuming large cardinals) that there is an extension of Le […]

I assume by $\aleph$ you mean $\mathfrak c$, the cardinality of the continuum. You can build $D$ by transfinite recursion: Well-order the continuum in type $\mathfrak c$. At stage $\alpha$ you add a point of $A_\alpha$ to your set, and one to its complement. You can always do this because at each stage fewer than $\mathfrak c$ many points have been selected. […]

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is negative. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${\mathfrak c}$ (This doesn't matter, all we need is that it is strictly larger. T […]

The power of a set is its cardinality. (As opposed to its power set, which is something else.) As you noticed in the comments, Kurepa trees are supposed to have countable levels, although just saying that a tree has size and height $\omega_1$ is not enough to conclude this, so the definition you quoted is incomplete as stated. Usually the convention is that […]

The key problem in the absence of the axiom of replacement is that there may be well-ordered sets $S$ that are too large in the sense that they are longer than any ordinal. In that case, the collection of ordinals isomorphic to an initial segment of $S$ would be the class of all ordinals, which is not a set. For example, with $\omega$ denoting as usual the f […]

R. Solovay proved that the provably $\mathbf\Delta^1_2$ sets are Lebesgue measurable (and have the property of Baire). A set $A$ is provably $\mathbf\Delta^1_2$ iff there is a real $a$, a $\Sigma^1_2$ formula $\phi(x,y)$ and a $\Pi^1_2$ formula $\psi(x,y)$ such that $A=\{t\mid \phi(t,a)\}=\{t\mid\psi(t,a)\}$, and $\mathsf{ZFC}$ proves that $\phi$ and $\psi$ […]

Yes, the suggested rearrangement converges to 0. This is a particular case of a result of Martin Ohm: For $p$ and $q$ positive integers rearrange the sequence $$\left(\frac{(−1)^{n-1}} n\right)_{n\ge 1} $$ by taking the ﬁrst $p$ positive terms, then the ﬁrst $q$ negative terms, then the next $p$ positive terms, then the next $q$ negative terms, and so on. Th […]

Yes, by the incompleteness theorem. An easy argument is to enumerate the sentences in the language of arithmetic. Assign to each node $\sigma $ of the tree $2^{

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.