Is it worthwhile to read GTM 53, Mathematical Logic for Mathematicians?

Question

I know little about mathematical logic. Graduate text in mathematics 53, Mathematical Logic for Mathematicians by Boris Zilber and Yuri I. Manin, looks appealing to me because it includes different aspects of logic such as theory of deduction, set theory, model theory, and alogrithms. (Of course not very deep in every area of logic, but I am not particularly interested in logic and want to understand it as quick as possible.) I also think it might be good because it is in the famous GTM.

My main concern is that the book may be too old to be worth reading. Although its second edition was published in 2008, a friend say that some of its notations and approaches are out of fashion. After all, its first edition is published in 1974! Is it a good book to read?

If there are better books, please tell me as well.

Answer 1

Some old books are definitely worth reading, and some new books are definitely not worth reading. But logic does have a particular issue that, as a relatively young field, many issues of notation and terminology are not standardized, and so older texts may indeed discuss things in a way that would not be found in the current literature. One of my favorite books, Monk's logic text GTM 37, suffers particularly from notation issues.

So let's look at this book by Manin and Zilber. The title page states that the first 8 chapters were translated from Russian by Neal Koblitz, so he must also have had some influence on the wording.

The sequence of topics is interesting, and you would certainly come away with some good knowledge. The topics include the basics of first-order logic, the incompleteness theorems, the independence of the continuum hypothesis, the MDRP theorem (which solved Hilbert's 10th problem), and some results from model theory. There are also some topics not usually included in this kind of book, such as quantum logic and quantum computing.

I don't think the book will bring you to a point where you could work in research in any of these areas. But as a general introduction, whether it's a worthwhile read is just a matter of taste. Glancing through several chapters, I didn't see any really unusual terminology or notation.

Answer 2

I have a book by Manin (without Zilber) titled "A Course in Mathematical Logic"; since it's also labeled GTM 53, I assume that the book you're referring to is an updated or expanded version of Manin's book, with Zilber providing the updating or expansion. I enjoyed reading Manin's book, partly because of some of the unorthodox topics. (What other logic book discusses Icelandic poetry?) I think the Manin book is a good one for logicians like me to read, because it gives a picture of our subject from the point of view of a first-rate mathematician who is not a logician.

Since I read Manin's book for my own enjoyment and enlightenment, without thinking of it as an introductory textbook, I didn't really form an opinion of its usefulness of the latter purpose. And I haven't seen the Manin-Zilber version, so I can't give you advice about that. (So this is not an answer, but it's too long to be a comment.)