Mathematical Logic

Rating:
Summary: Rock-solid introduction to Mathematical Logic
Review: Since my first contact with mathematical logic, I've always seen it as a kind of brainwashing, forcing one's mind to work based on several little pieces of thought. Nevertheless, it can be described as "a necessary evil", because the mindless use of mathematical logic throughout mathematics is very treacherous, as it can be seen in the problems regarding the axiom of choice, the Banach-Tarski paradox in measure theory, the issues about the undecidability of certain assumptions in set theory, and the very limitations of mathematical logic.

Usually, of course, most work in mathematics doesn't require a deep knowledge of rigorous mathematical logic, but it's always a good thing to a serious mathematician to have some acquaintance with it, even if it's just to avoid boobytraps. Then, it's hard to find a better choice than Shoenfield's book. After a long absence from the book market, A K Peters made the wise decision of reprint this masterpiece. Although most of its contents are fairly standard for a book on mathematical logic (unlike the equally marvellous out-of-print book of Yu. I. Manin, which has a more philosophical slant and concerns itself with issues such as quantum logic, literature, etc.), it provides proofs for many propositions that in most of the literature are only stated. It has, of course, some extras not generally found in other books, as for example issues concerning constructibility of sets.

But the most important characteristic of this book is its clarity and precision. It doesn't waste time in unnecessary stuff, and shows why we need mathamatical logic at all. Although it lacks some topics (for example, it doesn't discuss other axiomatic set theories besides Zermelo-Fraenkel. This is not so nice, because it lacks the distinction between classes and sets, one of the tenets of the Goedel- -Bernays-von Neumann set theory, although it is conceptually easier than this last one. But maybe it's a pedagogical choice, because the set theory we all intuitively know is more or less based in Zermelo-Fraenkel), its main concern is pedagogy, so this limitation has a sound reason: this book exposes mainly the logic present in the math most mathematicians and alike scientists (mathematical physicists, etc.) use. Its solidity and razor-sharp precision is great to instruct these people to be more careful with the math they use.

Besides that, some of the missing topics can be complemented by Mendelsson's "Introduction to Mathematical Logic", which is a bit more "merciful" book, which, by the other side, welcomes the thoroughness of Shoenfield.