A couple of weeks ago I busked a talk on ordinals and cardinals. Graham asked me to post a reference to a book.

One I learnt a lot from when I was young is

“The theory of sets and transfinite numbers”, B.Rotman and G.T.Kneebone, Oldbourne Mathematical Series, 1966.

My copy cost 35 shillings. I doubt very much its still obtainable – except that I still have my copy, and I’ll (try to remember to) bring it down when I’m back (19th July or so). There may well be better books to recommend nowadays, but … I haven’t read them.

The next time I give such a talk, to functional programmers, I think I’ll start off with the hierarchy of process types: data Proc2 = Stop2 | Beep2 Proc2 | Read2 (Nat -> Proc2) data Proc3 = Stop3 | Beep3 Proc3 | Read31 (Nat -> Proc3) | Read32 (Proc2 -> Proc3) ... Why the wierd “2″s and “3″s? Because Proc2 is the second number class (the countable ordinals), Proc3 is the third, and so on. Exercises: what’s the first number class? What’s the nought-th? Can you think of a way to write the omega-th in Haskell? In a different direction, is there something you are very familiar with that gives you addition? How’bout multiplication? Exponentiation?

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By the way, I recently attended “Computability in Europe 2007″ in Siena: http://www.amsta.leeds.ac.uk/~pmt6sbc/cie07.html. This is the first time I have attended such a large conference: about 285 participants. There were 4 parallel “special sessions”, and 7, yes 7 parallel streams of contributed talks. There were a lot of talks on biological computation. Educational. I came back with two breeze-block sized volumes of pre-conference proceedings that interested parties are welcome to peruse. Actually, Conor already has one: we should try to get Milli Maietti to come, because in some sense, she’s in the downstairs flat with Conor and Thorsten doing the OTT upstairs. (Poor woman…)

It was nice socially, but thin scientifically. (Apart from some remarks by Andrej Bauer.)