Mathematical Logic

by Heinz-Dieter Ebbinghaus, Wolfgang Thomas, Jörg Flum

Book Overview

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... This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 0387908951
• ISBN13: 9780387908953
• Release Date: January 1984
• Publisher: Springer
• Length: 216 Pages
• Weight: 1.10 lbs.

Customer Reviews

Excellent Choice for Teaching Mathematical Logic

This is a truly excellent book -- one I've used (along with other other books) to teach mathematical logic for 20 years. (The new edition provided welcome coverage of logic programming.) Traditionally, logic pedagogy has tended to revolve around which colleges or universities are involved. You will need to have sharp students to take full advantage of this textbook. In addition, some proof construction environment/proof checker is a good thing to have accompany the textbook; the same would hold of model finders. For grad students in my lab, I require familiarity with the book, sooner or later.

Reads like Mathematical Poetry

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.

An excellent book, but not for beginners.

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.

Very good *mathematical* logic book

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.

A super book!

For those who are mathematically inclined, this is the best upper-undergrad textbook introducing mathematical logic.