Two 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 just isn’t going to engender much understanding at all. Stick e.g. to Rautenberg.
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