I have been rather quiet here, not because life has been so wildly over-exciting as to distract me from blogging but for the opposite reason. Nowt much happening.
My Faculty office is, at the moment, out of bounds as the builders are in my end of the Raised Faculty Building stripping out and replacing the heating system. (“Why wasn’t it done when the building was refurbished not so many years ago?” Jolly good question. And next summer they are going to replace all the windows, so that will mean more disruption and another round of covering everything up and cleaning after … Brilliant money-saving efficiency, eh?) But an upshot is that I’ve found myself working in the Phil. Faculty library, for the first extended period in … what? … forty years?? Which all seems a bit strange. Though very quiet and congenial with no students around.
Anyway, as I’ve said here before, my plan was/is to write a book which says enough to explain what is going on in Gentzen-style proofs of the consistency of arithmetic (how do they work? what do they show given that we all think that PA is consistent anyway, don’t we?). But it isn’t going quite according to the original plan, as I found that there wasn’t anything that I could point to as giving the background theory of small countable ordinals in the way that I’d want to have it presented. There are aspects I really like about Wacław Sierpiński’s old Cardinal and Ordinal Numbers (from 1958); and the short treatment in my friend and colleague Thomas Forster’s Logic, Induction and Sets (CUP, 2003) is very helpful. But all the same, I find myself spending the time writing a long tutorial on small ordinals as the first part of the book as now re-planned. More precisely, this is a tutorial (addressed to myself as much as anyone!) on small-ordinals-without-set-theory. And as always with these things, doing it carefully with (I hope) total transparency about what depends on what takes for ever (or at least, takes me for ever).
Perforce, then, I’m taking it slowly. I just hope that when this part of the book is done someone else will find my route through this stuff an interesting one to take. Anyway, when it is gets to a natural division point and is sufficiently polished, I’ll make this tutorial available here (perhaps 120 book pages?) for no-doubt much-needed comments. Watch this space, though don’t hold your breath.
By the way, talking of Sierpiński, Rafal Urbaniak has pointed me to a whole list of books by Polish mathematicians which mentions another nine by him (gotta admire the energy!), including a seemingly rather nice book in English on the Theory of Numbers, available online .