## Kirby, An Invitation to Model Theory

I very briefly noted Jonathan Kirby’s An Invitation to Model Theory (CUP, 2019) when it was published earlier in the year. I put it aside to look at later, and (I confess) forgot about it entirely when recently updating the TYL Study Guide! Belatedly, I have now dipped into quite a lot of this short book; how does it compare as a first introduction to model theory?
The aim of the book is described like this: “[T]raditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, [the author] brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics ….” Now, one thing that usually isn’t familiar to undergraduate mathematicians is any serious logic: so, as you would expect, Kirby’s book is an introduction to model theory that doesn’t presuppose or seamlessly continue on from a first logic course. By contrast, in the TYL Guide, I suggest as a route into the area the long last chapter ‘Some use of compactness’ from Goldrei’s excellent logic text, and/or the more expansive but equally logic-based book by María Manzano.
The Invitation is divided into six parts, each comprising five or six short chapters (the main text of the book is only 176 pages long), with dependencies very well signalled — the book is a model of clear structuring. The first part ‘Languages and structures’ gives the logic-less student some basics about the ideas of a first-order language and interpretation in a structure, and then introduces the ideas of embeddings, substructures etc. The next part proves a compactness theorem and gives some first applications. Part III, ‘Changing models’ proves downward and upward Löwenheim-Skolem theorems, gives some applications, gives examples of theories which are countably categorical (or categorical in other cardinals), and in particular use. . .

News source: Logic Matters