Mathematical Logic (Undergraduate Texts in Mathematics) [Hardcover]

H.-D. Ebbinghaus (Author), J. Flum (Author), W. Thomas (Author)

Book Description

Publication Date: June 10, 1994 | ISBN-10: 0387942580 | ISBN-13: 978-0387942582 | Edition: 2nd

This introduction to first-order logic clearly works out the role of first-order logic in the foundations of mathematics, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines. It covers several advanced topics not commonly treated in introductory texts, such as Fraïssé's characterization of elementary equivalence, Lindström's theorem on the maximality of first-order logic, and the fundamentals of logic programming.

Editorial Reviews

Review

"...the book remains my text of choice for this type of material, and I highly recommend it to anyone teaching a first logic course at this level." - Journal of Symbolic Logic

Language Notes

Text: English (translation)
Original Language: German

Product Details

• Hardcover: 318 pages
• Publisher: Springer; 2nd edition (June 10, 1994)
• Language: English
• ISBN-10: 0387942580
• ISBN-13: 978-0387942582
• Product Dimensions: 9.3 x 6.3 x 0.7 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.7 out of 5 stars  See all reviews (12 customer reviews)
• Amazon Best Sellers Rank: #239,864 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

24 of 24 people found the following review helpful:

5.0 out of 5 stars Reads like Mathematical Poetry, December 14, 2003

By 

J. Fogel - See all my reviews
(REAL NAME)   

This review is from: Mathematical Logic: Undergraduate Texts in Mathematics (Hardcover)

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.

17 of 18 people found the following review helpful:

5.0 out of 5 stars Very good *mathematical* logic book, December 9, 1999

By 

Emre Domanic (Istanbul, TURKEY) - See all my reviews

This review is from: Mathematical Logic (Undergraduate Texts in Mathematics) (Hardcover)

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.

12 of 12 people found the following review helpful:

4.0 out of 5 stars Should be the standard undergrad introduction, July 11, 2005

By 

Nathan Oakes (Ashland, Oregon) - See all my reviews

This review is from: Mathematical Logic (Undergraduate Texts in Mathematics) (Hardcover)

Intended for a one-semester course, it ignores some of the usual topics in a survey course so it can give a deeper treatment of the nature and adequacy of mathematical proofs. It slights number theory, second-order logic, nonstandard analysis, and set theory. There is only enough on recursion and computability to support the main topic, but it goes deeper than usual on limitative results.

What it does cover it does very well. Motivation is rich and exercises follow well from the text. Proofs are very clear. Overall, there is much greater coherence in the development of ideas than you usually see in a survey text.

While the writing is very good, there is a shortage of definitions, examples, and exercises. Notation is not always clearly introduced and they adopt so many abbreviations it's hard to keep track of what things mean. I also thought that it was not as clear in the second half, maybe due to the multiple authors. Still, I would choose it over Enderton unless you need lots of exercises for class use.