Three shorter book notes
Yuri Manin who died last year was a seriously distinguished mathematician, being — for instance — one of the first recipients of the Schock Prize for mathematics. His interests ranged very widely, from algebra and topology to quantum field theory. So A Course in Mathematical Logic for Mathematicians (1977 translation, Springer) is written by an author with an usually broad background which interestingly shows through from time to time in the book.
The Preface starts: “This book is above all addressed to mathematicians. It is intended to be a textbook of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries of the last ten or fifteen years. These include: the independence of the continuum hypothesis, the Diophantine nature of enumerable sets, the impossibility of finding an algorithmic solution for one or two old problems.” So, yes, this is indeed aimed at pretty competent mathematicians. And yes, the attraction of the book when it appeared was that it gave early textbook accounts of e.g. Boolean valued models in set theory, or of the MRDP theorem.
Almost fifty years on, there are — quite unsurprisingly — more accessible accounts to be had of those then-much-more-recent results. And the level of discussion can be, well, ‘sophisticated’. So Manin’s book isn’t, whether in whole or in part, really a candidate for primary recommendations in the Beginning Mathematical Logic Study Guide. Still, it is distinctive enough that it should probably get a mention as being of possible interest to enthusiasts who find its style congenial. Indeed, I thought it did get just such a mention in earlier versions of the Guide; but it turns out that my memory plays me false.
A book I do mention in the current Guide is Volume II of George Tourlakis’s Lectures, i.e. the volume on set theory (I was probably unnecessarily polite about it!). But what about Volume I (CUP, 2003)?
This has two long chapters. The first (204 pp.) is called ‘Basic Logic’. We get an account of first-order languages, a Hilbert-style proof system (with all instances of tautologies as axioms), soundness, completeness and compactness proofs, and a little model theory (including e.g. the Loś-Vaught test and something about infinitesimals). Then we learn about computability and recursive functions, leading to semantic and syntactic versions of the first incompleteness theorem. The menu of topics is a standard one, then. But I just can’t bring myself to warm to the way the exposition goes, especially when it comes to Gödelian incompleteness. There are, in summary terms, much better options available.
The second chapter (116 pp.) is called ‘The Second Incompleteness Theorem’. The blurb advertises this as “the first presentation of a complete proof of Gödel’s second incompleteness theorem since Hilbert and Bernay’s [sic] Grundlagen” (but cf. Adamowicz/Zbierski 1997?). To my mind, Tourlakis’s presentation hacks through a mass of detail in a way that isn’t going to engender much understanding at all. Stick e.g. to Rautenberg.
The third book I’m going to quickly mention and then mostly pass by is Harrie de Swart’s Philosophical and Mathematical Logic (Springer, 2018). This really is a strange mish-mash of a book, supposedly aimed at philosophers. There are chapters (not very good) on e.g. the philosophy of language and “fallacies and unfair discussion methods”, and e.g. a section on social choice theory. Then there are chapters more directly relevant to the Guide on propositional and predicate logic, modal logic and intuitionistic logic.
If the overall structure of the book is a bit of a disorderly hodgepodge, so too can be the structure of individual chapters. Take, for example, the chapter on propositional logic. We get a bit of baby-logic level talk about truth-functions, then suddenly a compactness argument. Then we are back to baby-logic on defining validity. A digression on enthymemes is followed by a bit of metalogic. We get a messily-presented Hilbert-style proof system. Then there is a digression on an underdescribed natural deduction system. Next we get signed tableaux and a completeness proof (but who would guess from de Swart’s presentation that these can be made beautifully elegant)? The chapter finishes with a random walk through some paradoxes, and then some arm-waving history in a strange chronology. What a mess!
And so it goes through other chapters, I think. So I mostly can’t like this book (and looking again at what I briefly said in the Guide before, I was too kind). Except — though I’ll want to double-check this — I can probably still recommend the chapter on intuitionism which I thought before gives a nice enough account of one way of doing tableaux for intuitionistic logic.
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