Book note: Tent and Ziegler’s A Course in Model Theory

The CUP bookshop marked its 20th birthday with a day offering a big discount on top of the usual discounts, so I came away with a small pile of half-price logic books (I wasn’t enticed by anything on the philosophy shelves, but that’s another story).

One of them, the book I’d actually gone in to buy (not knowing about the birthday celebrations), was Katrin Tent and Martin Ziegler’s new contribution to the ASL ‘Lecture Notes in Logic’ series, A Course in Model Theory. To declare an interest, I’ve wondered more than once about writing a book on model theory useful for philosophers. Well, according to the blurb, ‘This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions.’ And according to the introduction, ‘This book aims to be an introduction to model theory which can be used without any background in logic. We start from scratch, introducing first-order logic, structures, languages etc. but move on fairly quickly to the fundamental results in model theory and stability theory.’

Well, for ‘fairly quickly’ read ‘very quickly’. In fact, I would have thought that a graduate student mathmo who really did have no background in logic would find this seriously tough going with stuff going by far too fast (what on earth would even a sophisticated such reader get out of the half a page on ultraproducts on p. 13?). And this quite certainly isn’t a book for graduate philosophers who want to learn some model theory.

It’s probably best, in fact, to think of it as something to read after having really mastered e.g. Wilfrid Hodges’s classic Shorter Model Theory, by way of consolidation and then extension. Taken in that spirit, it does indeed look useful for serious mathematicians (though I’m not going to have time to work through it further at the moment).  But this is probably not a book for beginners in model theory (and surely not one for graduate philosophers, even those of a logical bent).