Customer Reviews

Introduction to Mathematical Logic, Fourth Edition

5 star:		(4)
4 star:		(4)
3 star:		(0)
2 star:		(1)
1 star:		(1)

 

 

Nearly forty years after it was published (1964), Elliot Mendelson's Introduction To Mathematical Logic still remains the best textbook on the principal topics of this subject. Although the book does not presuppose any background in the subject or in any particular branch of mathematics, the reader should have some degree of "mathematical sophistication."

The first chapter starts with truth tables and ends with a completeness proof of a given formal system for propositional logic and an independence proof of the axioms of this system. Chapter Two is the study of quantification theory. Topics include quantificational completeness, Hilbert's Second Epsilon-Theorem, various topics from model theory, such as compactness and Lowenheim-Skolem Theorems, theorems on submodels and ultrafilters and non-standard analysis. The new fourth edition adds a very nice section on interpretations of quantification theory that allow the empty domain. Chapter Three presents an axiom system for number theory, recursive functions and proves (among other theorems) the famous Godel Incompleteness theorems, Tarski's indefinability of Truth Theorem and Church's Undecidability Theorem. Chapter Four is devoted to elementary set theory. Topics include an axiom system for set theory, ordinal and cardinal numbers, the axiom of choice and regularity, and alternative axiom systems of set theory. The new fourth edition includes an axiom system with urelements, something rarely presented, and an interesting note on the historical application of such a system in the construction of the first independence proof of the axiom of choice. The fifth chapter is the study of computability. The chapter begins with the notion of an algorithm and Turing Machines and builds up to the Kleene-Mostowski Hierarchy. The new fourth edition concludes with an excellent appendix on second-order logic.

I have used Mendelson's book to teach a one-semester course to advanced undergraduate and graduate students with great success. Such a course is centered on the first three chapters, omitting from Chapter Two anything beyond quantificational completeness. If time permits, I recommend either the rest of Chapter Two, the beginning of Chapter Five, or the appendix on second-order logic. Set theory, the content of Chapter Three, is usually offered as a separate course.

This book is a bit of an elegy to a dying world: the math logic of the 20th century.
It does not cover any nonclassical or philosophical logic, directions heavily researched in recent decades. Algebraic logic is slighted, even though Mendelson was an authority on Boolean algebra. Nor does he do justice to the model theoretic perspective, although the treatment of the Tarski semantics for first order logic in chpt. 2 is a bit of a classic. The treatment of recursion in chpts. 3 and 5 are thorough. The set theory of chpt. 4 is a bit unconventional (NBG rather than ZF) but is well exposited. My overall complaint is the crabbed notation, altho he's come a long way since the first edition. The book also cries out for a more graceful English style and page layout. Here Machover (1996) stands out.
Mendelson's bibliography is wonderfully long and rich. Finally, this text contains perhaps the gentlest extant introduction to second order logic.

I was sufficiently fortunate to have taken Professor Emeritus Mendelson's famous logic course at Queens College, the City University of New York, just two semesters before his retirement. I was, and continue to be, astonished by Dr. Mendelson's precise yet easy style, and the beautifully efficient organization of the subjects. Everything from the expository prose to the system of notational conventions has been carefully thought through so as to make the book both very substantive and very readable. In my opinion, it's the best introduction to serious mathematical logic currently on the market, and thanks to the genius of its author, it is likely to remain so for a long time. The buyer will not be disappointed.

This is one of the more popular introductory textbooks on mathematical logic, with Enderton's being its biggest competitor. I prefer Mendelson's for its breadth of material and the choice of proofs he uses, which are generally the most intuitive (e.g. Kalmar's for the completeness of the propositional calculus). This is not to say that they are always constructive, as they many of them are in the older texts (e.g. Kleene, Introduction to Metamathemaitcs).

The exercises are thoughtfully chosen. There's a good range of difficulty and a good portion of the answers can be found in the back. Difficult questions are indicated to the reader.

Out of all the mathematical logic texts I have (which are quite a few in number), this is the most oft-referred-to.

Mendelson's Introduction to Mathematical Logic was the textbook for a logic-course I took a couple of years ago. At the time I did not like the book at all. It seemed too difficult and so typographically ugly that I thought I would never use it. Things have changed though. Now, I keep it close at hand on my desk and use it almost every day. Technical questions that used to require a trip to the library and several different books to answer, can usually be resolved by a look in Mendelson's book. It's wonderfully rich and clear! I still don't find everything easy but that's because the material isn't easy and so not something Mendelson can be blamed for. I do find the typography ugly and at times annoying, but that's a small price to pay for a presentation as rigorous and detailed as Mendelson's.
So in summary: it's not the ideal book for the complete newcomer, but once you get past the initial hurdle it's a must read.

In my work as a math teacher, researcher, author and journal editor, I often encounter problems with a logical component. When that need arises, my first choice of reference is always this book. It is the most concise and readable introductory text I have ever encountered and it is a rare occasion when I fail to find the background material needed to solve the problem. It is also an excellent source of problems and I have pulled the ideas for many test questions from it over the years. Those problems have appeared on tests in both mathematics and computer science.
The topics are fairly standard, starting with the propositional calculus and covering quantification theory, formal number theory, axiomatic set theory and computability. What differentiates this book is the clarity of the description, making it ideal for an introductory course for undergraduates. Solutions to some of the problems are given in an appendix.
Logic is a fundamental component of mathematics and all mathematicians need some exposure to it. It is also a critical part of computer programming, as many sections of programs can be directly deconstructed into logical statements. Workers in both areas will find this book of enormous value.

This is a very useful and must have book for every graduate student in logic.Theory covers many fields(logic and computability) and has a lot of exercises (and also solutions to the tough ones)!!!

Even though I already knew the material, I found this book painfully slow to read. The author habitually writes in sentences that are runon, convoluted, repetitive, and indirect. I kept reading passages over and over to sort out what he was saying. That goes double for the proofs.

There is just not a clear unfolding of ideas at the sentence, paragraph, or chapter levels. It is even uninviting to look at; the layout is cramped and the notation is unnecessarily elaborate. The only point I can say in its favor is that it covers more material than most texts, as it is designed for a one-year course.

Late in August, the text originally selected for my mathematical logic class became unavailable. On the basis of reviews only, I chose Mendelson's Introduction to Mathematical Logic as the replacement. A disasterous choice. There may be a page without a typo, but I don't expect to find it. The presentation is inconsistent in notation and focus. Concepts are confused and more difficult than they should be. Definitions are not wisely selected. This book reads like something that has been patched for four decades (since 1964). On the positive side it contains interesting supporting material and will be a valuable private source of ideas to the lecturer. Be sure to read sections from chapter 2 and 3 before selecting this as a text.

Mendelson reaches an optimal point between the concision of the expert reference, and the wideness requested to a introductory text. Not in vain it has been the text forced in the universities during forty years.
Nevertheless, I believe to have found an error in the demonstration that does of the theorem of the completeness of the Predicate calculus, in the part in which it tries to demonstrate that all logical truth is
a theorem of the system.
[...]