Mathematical Logic (Undergraduate Texts in Mathematics)

Rating:
Summary: Reads like Mathematical Poetry
Review: As others have pointed out, this book is not for beginners, but is very well suited for those with some confidence in formal logic and axiomatized set theory. The book is just great if you want to deepen your understanding of the subject beyond what can be had from undergrad level courses on the topic. It should be required reading for any student of computational logic.

The question this book addresses is not "why logic?", or "what is a formal logic?", but more specifically, "why is first-order predicate calculus with equality such a good foundation for mathematics?"

The formal mathematics is organized and presented so clearly and precisely that I felt I was admiring a fine crystal structure.
The notation used may seem excessive to some, but it actually is the least amount of notation that could be gotten away with without resorting to glossing over fine distinctions. For example, many logic books assume a fixed countably infinite number of function and predicate symbols, which leads to some confusion when comparing different axiomatizations of the natural numbers, or of groups. This book on the other hand is crystal clear on how such different axiomatizations are related to each other. Another subtle point I never noticed before about first-order predicate logic but that is pointed out in the footnote on page 73 is that one might think it possible that just because a formula can be proven with one choice of predicate and function symbols, it might not be provable with a different choice of symbols. It turns out that this cannot happen as a simple consequence of the completeness theorem! (p. 85)

The book explores second-order predicate logic and makes explicit some of the difficulties, such as incompleteness and even the problem of how closely the truth of a formula in second order logic depends on what we take as true in set theory: different axiomatizations of set theory lead to different semantics for second-order predicate logic!

There is a great chapter on the incompleteness theorems, and in addition to Goedel's theorems, there is a section on Register Machines (a version of Turing Machines) and a proof of the undecidability of arithmetic using the halting problem, as well as a more general theorem about the undecidability of any theory that can encode the workings of a Register Machine.

The next section is a reasonable presentation of the mathematical underpinnings of logic programming.

The book concludes with an algebraic characterization of elementary equivalence followed by two deep theorems by Lindstrom that demonstrate the uniqueness of first order predicate calculus among formal languages with set theoretic semantics.

Rating:
Summary: Lots of typesets, and for what purpose?
Review: I do not recommend it as an introduction to mathematical logic.
I found the material to be insufficiently motivated. Unfortunately, the authors take some of their variables from an old english alphabet, which ruins the aesthetics. I am sure that some elegant and glorious principles are expounded here. But it was not written clearly enough for me to see them.

Rating:
Summary: Restrained, Serious, Comprehensive: A Revolutionary Textbook
Review: The best introduction to formal logic currently available, Ebbinghaus and Flum's *Mathematical Logic* drops the technique of "talking to cannibals" complained about by G.H. Hardy. Namely, Hardy couldn't do that anymore, and that was a long time ago: but for a long time readable logic texts (that is, texts not dedicated to the extrusion of logical categories from more copacetic algebraic ones) maintained a publicist's mien concerning the subject. Not so with this, but Ebbinghaus and Flum secretly maintain the publicist's attitude: this book covers more topics more thoroughly and elegantly than any of its competitors (including Wilfrid Hodges' much-touted, elsewhere-needed *A Shorter Model Theory*), including many topics of significance to graduate students in mathematics (infinitary logic, generalized quantifiers).

It really deserves to become the backbone of logic education -- that is, students deserve this and not half-hearted or whole-hog texts. For once, Springer is not blotting a superior product out of the market, but don't be confused: this is a rare occasion, as the serious aficionado of formal logic looks to North-Holland for guidance and Cambridge for avoidance. Not all "yellows" are equally mellow, and Springer's glut of nuvotexts is to be avoided like the intellectual plague it is -- many trends of the intellectual present cover up deeper perspectives with more appealing "localizations" not written all over them.

Rating:
Summary: Restrained, Serious, Comprehensive: A Revolutionary Textbook
Review: The best introduction to formal logic currently available, Ebbinghaus and Flum's *Mathematical Logic* drops the technique of "talking to cannibals" complained about by G.H. Hardy. Namely, Hardy couldn't do that anymore, and that was a long time ago: but for a long time readable logic texts (that is, texts not dedicated to the extrusion of logical categories from more copacetic algebraic ones) maintained a publicist's mien concerning the subject. Not so with this, but Ebbinghaus and Flum secretly maintain the publicist's attitude: this book covers more topics more thoroughly and elegantly than any of its competitors (including Wilfrid Hodges' much-touted, elsewhere-needed *A Shorter Model Theory*), including many topics of significance to graduate students in mathematics (infinitary logic, generalized quantifiers).

It really deserves to become the backbone of logic education -- that is, students deserve this and not half-hearted or whole-hog texts. For once, Springer is not blotting a superior product out of the market, but don't be confused: this is a rare occasion, as the serious aficionado of formal logic looks to North-Holland for guidance and Cambridge for avoidance. Not all "yellows" are equally mellow, and Springer's glut of nuvotexts is to be avoided like the intellectual plague it is -- many trends of the intellectual present cover up deeper perspectives with more appealing "localizations" not written all over them.

Rating:
Summary: Very good *mathematical* logic book
Review: This is *the* excellent mathematical logic book for anyone sufficiently familiar with the aims and spirit of mathematical logic. However, it is probably *not* suitable for a first introduction.

Some of the informal discussion expects the reader to supply the sense, and hence could be misleading for a novice (or even incorrect if taken literally!) On the other hand, the discussion is crystal clear and illuminating for someone with a bit more of background.

This book will not provide philosophical enlightenment to students of logic (esp. to those who seek such enlightenment in the first place), but it will provide good understanding of the study of general mathematical structures and their relation to logic. The prospective reader should first get acquainted with the model theoretic point of view (i.e. with its aims and presuppositions) before tackling this book. Good sources are: the first few chapters of Wilfrid Hodges's "A Shorter Model Theory" and the relevant articles by Jaakko Hintikka which were published in the journal "Synthese" in the late 1980's.

Rating:
Summary: A super book!
Review: This is an excellent book in its field. It is well-written, clear and presents insights as well as tools.

Rating:
Summary: An excellent book, but not for beginners.
Review: This is probably one of the best introductions to mathematical logic for those with sufficient mathematical maturity. I especially enjoyed the treatment of the completeness theorem for first-order logic (using Henkin's Theorem), and the treatment of Godel's incompleteness theorem, and Trachtenbrachts incompleteness theorem for second-order logic. Compared to other books, this book tends to go light on the notation.

If you do not have sufficient math maturity, then you may want to try Smullyan's book on the subject.