An accessible, very readable, well-motivated, zestful book on ordinal analysis and proof theory would be a very good thing to have. Arai’s Ordinal Analysis with an Introduction to Proof Theory isn’t it. By a country mile. (I was asked for a verdict: so here it is.)

Let’s ignore the fact that there are far too many garbled non-sentences such as “We denote instead of when no confusion likely occurs” (p. 2). Let’s pretend the book has been translated into a closer approximation to English. It may for all I know be packed from end to end with technical truths. But that is little recommendation if it remains dreadfully hard-going, a very paradigm of how not to write an enjoyable and attractively helpful introduction to its topic. Who on earth is it intended for? If a reasonably logic-literate reader like myself finds it rebarbative, what hope a graduate student wanting to learn some proof theory?

Not that this sort of thing is so very exceptional: it is depressing how badly written many maths books are. But you can certainly cross this one off your Christmas list.

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