## Ordinals

So I have seven lectures left in combinatorial game theory, and we’re probably 1.5-2 lectures away from being done with everything I want to cover in volume 1 (which is really pretty close to everything). I certainly hadn’t expected we’d move so quickly. Looking back at my proposed outline I made over winter break now just makes me laugh. I really underestimated…something. The students in the class? Myself? Some other factor? So I plan to spend a lecture on chess, and Dylan is going to give a lecture on go, but then there are still >=3 lectures left. So I’m thinking about talking about the material on infinite and loopy games from volume 2. (The other option would be to talk about misère games from papers by Aaron Siegel and Thane Plambeck, but it’s a bit more technical than I really want to get, especially for people who might not be particularly comfortable with group theory, much less the weird theory of commutative semigroups.)

So I think I can do the material on impartial loopy and infinite games without any trouble, but I’d like to get to the partizan games as well. And there isn’t much point in talking about those if students don’t understand ordinals. So I have to put together some material on ordinals. That’s fine with me, but I’m a bit lost for motivation. I looked through some “fun math” books I got when I was in high school/early college, and they all talk about cardinals, but none of them talk about ordinals. I tried writing up some notes on ordinals this morning, but they were clearly completely useless. Can anyone give me some inspiration for stuff to say about ordinals? Any books that treat them in a very elementary manner? Thanks!

I have all sorts of cool projects to do this quarter, but somehow they aren’t as inspirational to me as they ought to be. I gave a talk on homogeneous distributions in one of my classes a few weeks ago, and I’m supposed to give another talk on the Poisson summation formula and distributions and its number-theoretic applications soon (this is what happens when you’re a number theorist in a class of analysts…), but I don’t know how distributions come up in number theory, other than a paragraph on Herbrand distributions that I didn’t understand very well from Weil’s Basic Number Theory (which should never have been named that!).

Then I have a paper to write on quantum algorithms in algebraic number theory. I think this will be fun once I can get used to reading computer science papers. Reading computer science papers is a bit annoying because computer scientists are so concerned with efficiency. Of course, that’s most of the point of their subject, but it isn’t of mine, so it feels a bit weird.

And I was “accepted” (it feels funny to say I was accepted when I didn’t have to do anything to justify my competence, but they said they have plenty of room, so perhaps that’s why) to the algebra summer program. I’m excited about that!

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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### 5 Responses to Ordinals

1. I think that the first place I encountered ordinals was Rudy Rucker’s /Infinity and the Mind/, which I think qualifies as a ‘fun math’ book. My rigorous introduction to ordinals came mostly from Chapter 1 (preliminaries) of Munkres’s topology, although I don’t think that falls in to what you mean by “very elementary”? (also, have you seen my roommate’s notes for the combinatorial game theory class that was taught here last year? Again not sure they’re the right level of sophistication, but ordinals come in at lecture 18.)

2. kinejoshua says:

Conway gave a pretty good talk on ordinals every year at Mathcamp. I can’t remember much of it, other than that he motivated it with the idea of counting (so a (countable?) ordinal is a way to count the elements of a set) and he used these really brilliant fencepost diagrams to visualize countable ordinals and their arithmetic.
Good stuff. Maybe he has it written up somewhere?

• Try the last chapter of /The Book of Numbers/ by Conway and Guy? I guess that’s the other place I saw ordinals when I was younger. I remember the fencepost diagrams.

3. Yeah, Conway has a series of lectures on the Princeton math page, and he gives a really cool discussion of ordinals in his “Cantor’s Infinities” lecture. Here’s the link:
http://www.math.princeton.edu/facultypapers/Conway/
And congrats on the summer program? Are you going?

4. deviantq says:

Same with Infinity and the Mind for me. But I think being able to talk rigorously about “infinity + 1” (omega + 1) is a pretty good motivation, no? Maybe introduce with some material on order types, as it’s interesting to get the perspective of omega + 1 as a modification of how the < operation works, instead of in set-theoretic definitions.