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.

Sean FeeneyIn contrast to Aluffi’s book which is suggestive of a journey from the underground perhaps as far as Chapter 0, Hils and Loser’s book seems at first glance, despite the title, intended as something of a journey to nowhere according to the back jacket: “The aim of this book is to present mathematical logic to students who are interested in what this field is but have no intention of specializing in it. The point of view is to treat logic on an equal footing to any other topic in the mathematical curriculum.”

An essential part of the exposition, according to the introduction, is the exercises: “We consider exercises as an essential component of the book, and we encourage the reader to work them out thoroughly; they should be seen not only as a tool to check that the course is correctly assimilated, but also as a way to provide an opening to additional topics of interest.”

So, the book is not necessarily a journey to nowhere; it could be final destination unknown.

And so there is rather more interest in the exercises than Peter’s parenthetic aside suggests, since the answer the authors presumably intended to the rhetorical question he poses is: a mathematical student or a mathematician who prefers a presentation that requires working through (presumably many) mathematical exercises, rather than explicit conceptual exposition.

The authors presumably think others also believe this is a worthwhile presentation, since the Acknowledgments name those “who encouraged us in the project of transforming our notes into a book ”

The introduction suggests in the preceding sentence what those “additional topics” might be: “In each chapter we have tried to present at least a few juicy highlights, outside Logic whenever possible, either in the main text, or as exercises or appendices.”

Three possible destinations are suggested after all in the suggestions for further reading at the end of the introduction: model theory, set theory, and recursion theory.

Nine of the thirteen books in the Bibliography are suggested as “providing more comprehensive and advanced material” for “the interested reader” – an important subset of the target audience is someone for whom this will actually be their first journey into logic.

So a condition for success of the book is that it would adequately prepare an “interested reader”, from its intended mathematical readership, student or mathematician, for reading each of the named nine books, if such a reader has worked the exercises “out thoroughly”.

https://www.ams.org/books/stml/089/stml089-endmatter.pdf

Sean FeeneyRather than concentrate on the subtitle does the title not predicate that A Second Journey is under contemplation?

Sean FeeneyIf it is an allusion to Dostoyevsky, it’s possible (perhaps even definite given the differences in titling) that some kind of hint of a journey towards category theory is the metaphor.

Rowsety MoidThough it might not make much difference to your comments,

A First Journeydoes say it’s intended for “advanced undergraduate students” (and up), rather than plain “undergraduate students” (and up). That’s the sort of thing a graduate-level text might say, and that’s what I think it is. So (despite the title) not a text for beginners, though (depending on how it handles the 2nd incompleteness theorem?) it might have a role as supplementary material.*

I don’t quite share your enthusiasm for

Notes from the Underground.1. It’s rings-first. (Having suffered through a rings-first course as an undergraduate, I wouldn’t wish it on anyone.)

2. It’s rings-first because Aluffi thinks groups require too much abstraction to be the start of a gentle introduction. And the book does indeed start quite gently, not reaching even rings until chapter 3. Then he seems to decide abstraction isn’t much of a barrier after all and goes from rings to the category of rings and even through modules (“and Abelian groups”) before eventually reaching “Groups — Preliminaries” (preliminaties!) as Chapter 11. Which begins 11.1 “Groups and their Category”. (Of course it does!) For the part on fields, he says the audience is expected to “be comfortable handling abstract concepts, perhaps at the level of a Master’s student”.

So not such a gentle introduction after all.

Why so long a route to groups, even making them secondary to modules? (Abelian groups are introduced as Z-modules; groups eventually appear as a modification of abelian groups.) An underlying reason may be that if a goal is for the reader to “emerge” with “a natural predisposition for absorbing the more sophisticated language of categories whenever the opportunity arises”, then the simplicity of groups is a distraction. Better that the reader see groups as just one thing among many, with categories as a simple, “unifying” context. Otherwise students might notice that groups are quite fun and wonder why so much abstraction is being piled on top.

2. Why is it called

Notes from the Underground, an allusion to Dostoyevsky? This is nowhere explained. It’s hard to escape the suspicion that it’s to make the book sound a bit edgy and cool. (The publisher’s blurb says it “follows an unconventional path.”)The book is curiously coy about category theory. Despite occasionally mentioning that it “underlies everything” in the book, or is the reason for certain decisions, even saying it is “possible, but

harder” to do without mentioning the concept (really?), he has “even refrained from giving the definition of a category”. This seems to be a tactic to entice the reader into finding the definition for themself: the first exercise is to “Read (or at least skim through) the Wikipedia page on ‘category theory’.”3. How much abstract algebra does someone interested in mathematical logic need to know? Not very much, it seems to me, certainly less that what’s in this book, especially since it’s over 200 pages before he finally relents and says what a group is.

Of course, some people are quite keen on abstract algebra, and on category theory. This would be a good book for them (and it does seem to be well-written). The reader has to be pretty darn keen, though, in my opinion.

Peter SmithA propos of Aluffi, I don’t think someone interested in mathematical logic needs to know much algebra. My interest, such as it is, is independent of that (and more philosophy of mathsy). I haven’t read enough yet to know whether I agree with your view about best-order-of-topics!

Rowsety MoidAnother reason for giving priority to rings may be the role they play in certain other areas. For instance, a scheme is

Put another way, one reason for thinking rings are a natural, useful thing that would make a good starting point may be the great familiarity with rings that comes from working in such areas.

(One thing I like about the

Undergroundbook, btw, is that it doesn’t — so far as I’ve been able to tell — to make the sort of ideological attack on set theory that some category theorists like to make.)