The authors write that they have “deliberately chosen not to write another comprehensive textbook, of which there already exist quite a few excellent ones, but instead to deliver a slim text which provides direct routes to some significant results of general interest.” But they also say that the book is intended for “undergraduate students, graduate students at any stage, or working mathematicians, who seek a first exposure to core material of mathematical logic and some of its applications.” So we are led to expect the chapters here to be accessible to beginners on a topic, though perhaps beginners with a reasonable level of mathematical maturity.

So how do things go? And more specifically — my immediate interest, of course, in looking at this book here and now! — are any of the chapters appropriate for recommending in the *Beginning Mathematical Logic* Study Guide?

Chapter 1 is called “Counting to Infinity”, and is more-or-less entirely a fast-track introduction to ordinals and then cardinals treated set-theoretically but informally (i.e. without getting entangled with formalized ZFC). Before the exercises start, this is just 21 relentless pages of definitions/theorems/proofs with very little by way of explanatory chat. To take just one example, the naive reader may very well wonder why ordinal exponentiation is defined in terms of functions with finite support (a notion which is defined, used once, and never motivated). What kind of reader is going to find this sort of presentation helpful? Perhaps as after-the-event lecture notes, following on from more informal presentations, these pages might have been useful to those attending the course from which these notes arise. But as a stand-alone text, the body of this chapter has nothing to really recommend it. (There is some interest in the exercises though.)

Chapter 2, “First-order Logic” is if anything worse. It is a baldly presented gallop through a Mendelson-style axiomatic system (with particularly dense thickets of symbols along the way). I really can’t imagine anyone coming away from, e.g., the completeness proof with a good sense of what’s going on: there are so many, much better, treatments out there.

Chapter 3 is “First Steps in Model Theory”. We get Tarski-Vaught after 3 pages (compare, after 43 pages of Kossak’s *Model Theory for Beginners*, and after 66 pages of Kirby’s *An Invitation to Model Theory*), and we get quantifier elimination after 8 pages (not treated by Kosssak, and Kirby takes 97 pages to introduce the idea). We get to Ax’s Theorem in 16 pages. I wonder how many will be illuminated by this sort of ultra-rushed tour?

Chapter 4 is on “Recursive Functions”, defining primitive recursive, partial recursive and total recursive functions, touching on Turing machines (without a single diagram or a single program illustrated), proves the Turing-computable functions are recursive, proves the unsolvability of the halting problem, etc. Now, this is markedly less sophisticated material than the model theory in the previous chapter, so this present chapter is probably quite a bit more accessible. But on the other hand, there is zero elegance here, and yet this is an area where we can make the concepts and the proof-ideas look so delightful (and thereby engender a good understanding). So I again can’t recommend this as a good read.

Sorry to be, thus far, so consistently negative: but take it as a community service to warn off any unwary readers. And I would have given up on the book at this point, except that the next chapter on arithmetic promises “a complete proof of Gödel’s Second Incompleteness Theorem”. *Really*? I think I should look to see …

And just to balance out all that negativity, let me offer some brief but very warm praise for another book (of a rather different kind). I couldn’t resist picking up a copy of Paolo Aluffi’s *Algebra: Notes from the Underground* from the CUP Bookshop a couple of days ago. This is so far looking terrific, a real pleasure to read. It is “only” an undergrad intro to algebra (I have my reasons for wanting to look at an up-to-date entry-level text to see how it handles things). But, as with his big grad-level book, I do very much admire and envy Aluffi’s presentational style. I may say more about this when I have read more.