Book Description
The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them. Introduction to Mathematical Logic includes: · propositional logic · first-order logic · first-order number theory and the incompleteness and undecidability theorems of Gödel, Rosser, Church, and Tarski · axiomatic set theory · theory of computability The study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields. ... Read more

Customer Reviews (10)

A must have....
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)!!!

twisted pants unleashed on men
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.

Wonderful at the second glance.
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.

Best reference in first step math logic
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.
[...]

Not a good read
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. ... Read more

Book Description

Customer Reviews (1)

Oldie But Goodie
I first read this book in 1972 for a college course in the Philosophy of Mathematics. I was a philosophy major, mainly because I thought I was no good at math. By the end of the first chapter, I understood geometry for the first time. By the end of the second chapter, I was hooked. Mathematics has been a favorite hobby and occupation ever since.

Unfortunately, my prized copy of this text was lost in a flood in 1982. So I am delighted that this reprint is available. If you enjoy math puzzles, if you are interested in Gödel or non-Euclidian geometry, or if you simply want to stretch your mind a bit, I heartily recommend this book. ... Read more

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Customer Reviews (3)

No-nonsense survey of logic
This is an introduction to the main ideas and results of mathematical logic. It is primarily a text for non-logicians but it is still very serious. Practically everything is proved, and the proofs are carefully crafted and not too technical. For a reader with a bit of mathematical background this is far more valuable than the more typical logic-for-casual-readers books, such as for instance "Gödel's Proof" by Nagel & Newman, which are too chatty and trivial and don't really prove anything. By contrast, a high point of this book is a very accessible treatment of the proof of Gödel's incompleteness theorem in a matter of a few pages. On the other hand, this book is perhaps not chatty enough: the clear proofs and discussions of the main results are nicely done, but the discussions of historical background, motivation and context are very sketchy.

Six Rigorous Lectures - Not for the Faint-Hearted
Although this book - What is Mathematical Logic? - is written in an informal and entertaining style, it is unlikely to appeal to a reader not familiar with predicate calculus, recursive functions, and set theory. Despite its innocuous title, this little book is surprisingly rigorous.

The six chapters are derived from a series of lectures given by the five authors - J. N. Crossley, C. J. Ash, C. J. Brickhill, J. C. Stillwell, and N. H. Williams - at Monash University and University of Melbourne in 1971.The lectures were substantially revised for publication.

Only the first chapter, a detailed historical survey of mathematical logic, can be readily appreciated by the non-mathematician. The remaining five chapters examine advanced topics in mathematical logic including the Godel-Henkin Completeness Theorem, Model Theory, Turing machines and recursive functions, Godel's Incompleteness Theorem, and advanced set theory.

Chapter 2 introduces the Godel-Henkin Completeness Theorem, a proof that predicate calculus is complete. Chapter 2 is not easy, but it is essential to acquire a reasonable familiarity with predicate calculus before moving forward.

Chapter 3 offers a detailed look at model theory, the study of relations between formal languages and the interpretation of formal languages. Topics include Predicate Calculus with Identity, the Compactness Theorem, and the Lowenheim-Skolem Theorems. I had substantial difficulty with the details, but I did gain a general understanding and appreciation for model theory.

Chapter 4 addressed in considerable detail a more familiar topic, Turing machines and recursive functions. The discussion concludes with a key proof: there is no algorithm which will enable us to decide, given any particular formula of predicate calculus, whether or not this particular formula is deducible from the axioms of predicate calculus.

Chapter 5 was a detailed examination of Godel's Incompleteness Theorem for formal systems that include arithmetic of the natural numbers. I had less difficulty with this topic as I had previously read Godel's Proof by E. Nagel and J. R. Newman. This chapter would very likely be tough going for a reader entirely new to Godel's exceeding complex and abstruse proof.

Chapter 6, titled Set Theory, might be better named Advanced Set Theory. I was entirely new to the Axiom of Choice and the Generalized Continuum Hypothesis.

I highly recommend this intriguing and lively look at mathematical logic to readers with some familiarity with this rather formidable subject. For readers new to mathematical logic, I suggest that the following books might be better starting points.

Foundations and Fundamental Concepts of Mathematics by Howard Eves is outstanding. The chapter titled Logic and Philosophy is an excellent introduction to mathematical logic.

The Advent of the Algorithm by David Berlinski is an eclectic, rather bizarre introduction to a complex mathematical topic. Although many reader reviewers aggressively criticize this book, I enjoyed puzzling my way through Berlinski's discursive discussions.

Godel's Proof by Ernest Nagel and James R. Newman offers a fascinating look at a mind boggling, incredibly complex, inventive mathematical proof.

Dense but readable
After a 10-page historical survey of logic from the 1850s through the 1960s, similarly brief chapters on Completeness, Model Theory, Recursion Theory, the Incompleteness Theorems, and Set Theory give an idea of what might be covered in an undergraduate course and the first several graduate courses in mathematical logic. (The last 5 pages of the book are an introduction to forcing arguments and a fairly detailed sketch of the consistency of not-GCH.)

Results are clearly and carefully stated; and while sketches of proofs have a hard time staying nontechnical and still meaningful, most such attempts are admirable.

A marvel of brevity while not watering anything down. ... Read more

Book Description
...classic introduction to the main areas of mathematicallogic......provides the basis for the first graduate course in thesubject. ... Read more

Customer Reviews (2)

Standard by default
Almost four decades after being written, this is still the standard graduate survey text. A large part of the reason is that there is little competition, but it is also a good book on its own merits. The author writes with a clarity and concision you rarely see in a math (or any) textbook. Proofs are straighforward, not tricky or convoluted. There are many excercises and with detailed setup. The exercises are often quite hard, requiring significant extension from the text.

Although the writing is good, that doesn't mean it is easy. He progresses deliberately through the details, rarely giving an overview. I think he is just expecting that you already have a good sense of context from the undergrad logic course you took (didn't you?). Sometimes he seems to belabor a point. There is also a dearth of examples, just five in the whole book, three of them in the appendix. There are no references at all. The age of the book makes it, not wrong, but inadequate in some areas. Still, I have looked at alternatives and haven't found something better for a graduate survey text in English.

Rock-solid introduction to Mathematical Logic
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 inmeasure 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. Itdoesn't waste time in unnecessary stuff, and shows why we need mathamaticallogic at all. Although it lacks some topics (for example, it doesn't discuss otheraxiomatic set theories besides Zermelo-Fraenkel. This is not so nice, because itlacks the distinction between classes and sets, one of the tenets of the Goedel- -Bernays-von Neumann set theory, although it is conceptually easier than thislast one. But maybe it's a pedagogical choice, because the set theory we allintuitively know is more or less based in Zermelo-Fraenkel), its main concern ispedagogy, so this limitation has a sound reason: this book exposes mainly the logic present in the math most mathematicians and alike scientists (mathematicalphysicists, etc.) use. Its solidity and razor-sharp precision is great to instruct thesepeople 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. ... Read more

Book Description

Customer Reviews (9)

Good for warming up your brain.
Nice collection of puzzles with varying difficulties, which do not require any special knowledge of mathematics.

The best compilation from Martin Gardner's Scientific American mathematical games column
"My best mathematical and logic puzzles" presents 70 of the best of the brain teaser that Martin Gardner published over a period of 25 years in his Mathematical games column at Scientific American. It some cases references to new developments related with specific puzzles have been added.

Martin Gardner was always especially careful to present in his American Scientific column only new and unfamiliar puzzles that have not been included in classic collections before. Now you can challenge your solving skills and rattle your ego with a compilation of his best mind-benders.

Here is an example of what you can find inside this book (31. The absent-minded teller}:

"An absent-minded bank teller switched the dollars and cents whenhe cashed a check for Mr. Brown, giving him dollars instead of cents, and cents instead of dollars. After buying a five-cent newspaper, Mr. Brown discovered that he had left exactly twice as much as his original check. What was the amount of the check?"

One of the best things about Martin Gardner books is that a carefully explained solution follows each problem, this way you learn and add new abilities to your problem solving skills, that will sure be helpful in solving real life problems, while entertaining yourself with a good and challenging reading.

The best of one of the best
Martin Gardner is the grand old man of puzzles and recreational mathematics. I recommend this book for intermediary and advanced puzzle enthusiasts - beginners might find some of these too challenging.

Intermediary puzzlists will find the pleasures of often working at the upper edge of their skills. The solutions at the end of the book are complete enough so that even those who didn't get it right the first time will get aha insights.

The book is well worth its price even for puzzle enthusiasts. Even I knew many of the puzzles beforehand - classics indeed - but the notes in the solutions often add a twist, a clever solution or a human interest point of view.

The age recommendation of amazon.com - 4-8 years - is probably either an insider joke or a typo. I'd recommend this book to people between 14-80 years of age, and even over.

Puzzles requiring intermediate mathematical skills
Marvelous book. I found it better than many books but my friends, who were not that conversant with intermediate mathematics did not like it much. Though this book doesn't require a knowledge of calculus, people who have this level might appreciate the book more. But it has more to do with mathematical 'thinking' rather than mathematics itself.

So get this one if you are good at mathematical thinking and want to challenge yourself. If you are weak in math and would rather read puzzles that require only logic, cleverness, and lateral thinking only, this may not be the one for you.

A Question
I just had to question this - the book is rated at a reading level for 4 - 8 year olds, but some of the reviewers mention going on technicalinterviews and keeping track of columns in Scientific America over a numberof years? ... Read more

Book Description

Customer Reviews (2)

Wide Ranging Look at Advanced Topics - See Alternative Textbooks Below
In 1977 Hao Wang gave a series ofadvanced lectures on mathematical logic to the Chinese Academy of Science.Wang's presentations were well-received and subsequently published in 1981 under the title Popular Lectures on Mathematical Logic. Wang's lectures are not intended for the layman.

A 1993 reprint edition by Dover includes a useful postscript in which Wang briefly outlines recent advances in mathematical logic.

In his introduction Wang states: The reader is advised not to spend too much time on any particular point that seems to present serious difficulties. It is to be expected that various readers will find certain parts too elementary or too advanced.

I found very little that was too elementary. Wang's extensive appendix (more than 100 pages) should perhaps be scanned before reading the text itself.

Setting the difficulty aside, Wang's lectures are quite good. He carefully traces the historical development of key ideas, develops a unifying framework, and identifies opportunities for new work. The computer sections are now outdated, but still make good reading from a historical perspective.Wang's lectures can also serve as a valuable reference source.

The primary sections are titled One Hundred Years of Mathematical Logic, Computers, First Order Logic, Computation: Theoretical and Practical, How Many Points on a Line?, and Unifications and Diversifications.Example topics include Godel's Incompleteness Theorems, Model Theory, the Lowenheim-Skolem theorem, Ramsey's theorem and indiscernibles, and Cantor's continuum hypothesis (GCH).

Looking for something less daunting?Foundations and Fundamental Concepts of Mathematics by Howard Eves offers several excellent introductory chapters on mathematical logic,symbolic logic, and axiomatic set theory. Five stars.

An Introduction to Symbolic Logic by Susanne Langer is a bit dated, but it is a good place to start. Four stars.

Godel's Proof by Nagel and Newman is an exceptionally good introduction to Godel's remarkable work. Five stars.

The difficulty level of What is Mathematical Logic? by J. N. Crossley (and others) is an advanced text that warrants the extra effort required. Five stars.

Introduction to Mathematical Philosophy by Bertrand Russell is the classic introduction to logicism. Will require careful reading. Four stars.

The Advent of the Algorithm by David Berlinski is a strange, eclectic, discursive, popular work. Not a textbook. Three stars.

Good Unifying View
Although this book may be somewhat outdated (published in the mid-70's), it does provide a cohesive view of the developments in logic up until that point. One gets a very strong sense of the status of logical development, while at the same time receiving a historical motivation for the methods employed in developing the theory. Many proofs are shortened or synopsized, however the integrity and technical level of the work is never compromised. In my opinion, the sections on Model Theory, Set Theory, Proof Theory and Recursion Theory provided the reader with a good sense of the major results in those areas. The section on computers (and their limitations) was a hoot to read, because of the limited view provided by the author, but otherwise, Wang has a strong intuition as to where modern developments could have led. Recommended for anyone trying to get a unifying view of the major developments in logic. ... Read more

Book Description

This textbook is written for advanced undegraduates or first year graduate students of mathematics and computer science. It is also intended for the working mathematician who wants to gain an appreciation of Mathematical Logic. There are no prerequisites for this book; however, some mathematical maturity is required. The book is written in a totally mathematical style, and any mathematician should feel at home reading this. The book starts with the definition of first order languages, proceeds through propositional logic, completeness theorems, and finally the two Incompleteness Theorems of Godel. In the process, the reader is also introduced to model theory and recursion Theory. After reading this book, the reader will be ready to branch into model Theory, recursion Theory, axiomatic set theory or even theoretical computer science.

Book Description

Customer Reviews (2)

AGood Mathematical Logic Text
This book is suitable foradvanced undergradutes and graduate students for learning mathematical logic.It contains a wide selection of exercises. In the back of the book, the author gave answers to selected exercises. But I think the notational conventions is a little of old, not stylish. I don't like this kind of notational convensions in the book.

I've worked throught the first two chapters
There are 5 chapters in total, of which I've worked through the first two on my own. These take you clearly through the metatheory of propositional and predicate calculus. I'm not a mathematician, so I was looking for something clear. I got this from the library along with several others including Enderton and Mendelsohn, and this was the one I ended up working with because it struck me as the clearest. Robbin does a good job of getting to the essentials without getting bogged down in a mass of theorem proving, but at the same time doesn't skip anything that you really need to know. This is a good, fast intro to the subject. ... Read more

Book Description
The foundations of mathematics include mathematical logic, set theory, recursion theory, model theory, and Gödel's incompleteness theorems. Professor Wolf provides here a guide that any interested reader with some post-calculus experience in mathematics can read, enjoy, and learn from. It could also serve as a textbook for courses in the foundations of mathematics, at the undergraduate or graduate level. The book is deliberately less structured and more user-friendly than standard texts on foundations, so will also be attractive to those outside the classroom environment wanting to learn about the subject. ... Read more

Book Description

Mathematical Logic for Computer Science is a mathematics textbook with theorems and proofs, but the choice of topics has been guided by the needs of computer science students. The method of semantic tableaux provides an elegant way to teach logic that is both theoretically sound and yet sufficiently elementary for undergraduates. To provide a balanced treatment of logic, tableaux are related to deductive proof systems.
The logical systems presented are:
- Propositional calculus (including binary decision diagrams);
- Predicate calculus;
- Resolution;
- Hoare logic;
- Z;
- Temporal logic.
Answers to exercises (for instructors only) as well as Prolog source code for algorithms may be found via the Springer London web site: http://www.springer.com/978-1-85233-319-5

Book Description

This volume offers insights into the development of mathematical logic over the last century. Arising from a special session of the history of logic at an American Mathematical Society meeting, the chapters explore technical innovations, the philosophical consequences of work during the period, and the historical and social context in which the logicians worked.

Book Description
This book is a text of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries of the last 10 to 15 years, including the independence of the continuum hypothesis, the Diophantine nature of enumerable sets and the impossibility of finding an algorithmic solution for certain problems. The book contains the first textbook presentation of Matijasevic's result. The central notions are provability and computability; the emphasis of the presentation is on aspects of the theory which are of interest to the working mathematician. Many of the approaches and topics covered are not standard parts of logic courses; they include a discussion of the logic of quantum mechanics, Goedel's constructible sets as a sub-class of von Neumann's universe, the Kolmogorov theory of complexity. Feferman's theorem on Goedel formulas as axioms and Highman's theorem on groups defined by enumerable sets of generators and relations. A number of informal digressions concerned with psychology, linguistics, and common sense logic should interest students of the philosophy of science or the humanities. ... Read more

Product Description
This introductory graduate text covers modern mathematical logic from propositional, first-order, higher-order and infinitary logic and Gödel’s Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory.

Based on the author’s more than 35 years of teaching experience, the book develops students’ intuition by presenting complex ideas in the simplest context for which they make sense. He also provides extensive introductions to set theory, model theory and recursion (computability) theory, which allows this book to be used as a classroom text, for self-study, and as a reference on the state ofmodern logic. ... Read more

Customer Reviews (3)

Where's the proof?
Quoting the author Hinman (page xi):"A notable lacuna is Proof Theory,
which fails to appear largely due to the incompetence of the author in this area".
And he is correct: there is no proof theory in this book, no Hilbert axioms,
no Gentzen natural deduction, nothing.So, on the one hand, we have the largest logic
book I have ever seen -- and yet, ironically, the most incomplete.
I'm sure there's a lot of good stuff in this book and it is written well but it's
missing half the story, i.e., it is missing an exposition of how one manipulates
symbols formally to prove theorems. Even the semantic side is incomplete:
there's no semantic tableaux, no resolution.So, this is not a good introduction
to logic.
By far the best introduction to logic I've found is "Mathematical Logic for
Computer Science" by Mordechai Ben-Ari.Serious/pure mathematicians of course will
want to continue with the likes of "An Introduction To Mathematical Logic"
by Elliott Mendelson.

A Comprehensive Graduate Text
If I were a young graduate student in mathematics looking for that one "perfect" graduate text on mathematical logic to purchase with my (very) limited income, I would buy a copy of Professor Hinman's book.In just under 900 pages, Hinman provides an extremely well written and informedintroduction to propositional logic, first order mathematical logic, axiomatic set theory, model theory, and recursion theory.Indeed, the book is written so well that a motivated student with the requisite background can easily profit from independent study---a statement that simply cannot be made about many of the other "classic" references in this difficult field.One great virtue of having a single reference that introduces these diverse but interconnected areas is the uniformity of notation and definitions;the reader need not pull his hair out cross-referencing between texts that use wildly different notation and, occasionally, different definitions.

I studied mathematical logic at the University of Colorado--Boulder in the late 1970s.In those days, the logic students all depended on a standard list of references to prepare for the PhD qualifying examinations, and it is significant that all or nearly all of those works are still in print.At the introductory level we read the magnificent books on mathematical logic and set theory by Herbert Enderton.At the graduate level, we read Shoenfield, Monk, Mendelson, and Manin for mathematical logic, Chang and Keisler for Model Theory, Jech (and to a lesser extent, Kunen) for set theory, and Hartley Rogers for recursive function theory.In the course of plodding through these references, I discovered a wonderful comprehensive text by John Bell and Moshe Machover and quickly elevated it to primary status on my reading list.Bell and Machover remains my favorite among the older references today, nearly thirty years later, both in terms of comprehensive coverage and clarity of prose;when I reach for a reference to clarify an issue onfoundations, Bell and Machover is the first book I turn to.

The new book by Hinman achieves the same comprehensive goals of Bell and Machover, providing a rigorous and coordinated introduction to logic, set theory, recursion theory and model theory.However, Hinman incorporates some research topics that have emerged in the years since the 1977 publication of Bell and Machover, and it includes some more traditional topics that were difficult to find in the earlier texts.To give one example, Hinman provides a brief introduction tothe axiom of determinacy. This topic was made available to non-specialists in two papers published in the AMS Notices of June and July, 2001, where Hugh Woodin of Berkeley discussed the axiom of projective determinacy and other hypotheses within the context of possible enlargements of ZFC that would resolve Cantor's famous continuum hypothesis.A second example is Hinman's very lucid treatment of forcing;this writer has always had difficulty understanding the very few presentations of Paul Cohen's forcing technique that have been available in the older texts, but I found Hinman's treatment exceptionally clear and easy to follow.

Professor Hinman states that this book resulted from his nearly 40 years of experience teaching mathematical logic to graduates and undergraduates.The truth of this claim is reflected in the exceptional clarity of the prose and the coherence as one skims across different chapters.It is apparent that serious thought, consideration for the reader, and years of experience in the classroom shaped the final form of this text.Given the paucity of new texts in mathematical logic and foundations, the publication of this book is truly a cause for celebration.If you can only afford one text on the subject, purchase this one;if you are burdened with an abundance of spare change, I recommend buying Bell and Machover as a second reference to supplement Hinman.

Introduction to Mathematical Logic
This is possibly the best book on general mathematical logic at the graduate level. ... Read more

Book Description
A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional coverage of introductory material such as sets.

* Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses.
* Reduced mathematical rigour to fit the needs of undergraduate students ... Read more

Customer Reviews (9)

John Wilson
Keen students may find if they study and parse both editions of Enderton's

Logic they may find much of interest. Getting to the root of a problem

can be of use in many situations. So best of luck.

Moderately difficult and very effective
This is the most clear book on intermediate level logic that is available.I have many of the logic books that are on its level, and this one is perfect.It covers the most important, difficult concepts in the easiest way possible.It is above all clear (though very terse).It is easier than Mendelson's text but, in my opinion, as it pertains to First Order Logic and Computability Theory, one learns no more through Mendelson's approach.

Perhaps its only problem is that it might be just a bit too difficult without an understanding, helpful instructor (or TA) to guide one through the exercises.At any rate an effective progression up to the book might entail: Patty's "Foundations of Higher Mathematics", to Klenk's "Understanding Symbolic Logic", to "Logic, Sets, and Recursion" by Causey.Only after equivalent material has been understood thoroughly can the more hardcore semantics and mathematics of Enderton's book be fully comprehended.And, gone at alone on one's free time such a progression might take up to 2.5 years, maybe more.

Readable but a bit rough
It tries to be a readable undergrad introduction and mostly succeeds. Explanations are generally not tight and memorable, proofs seem loose, there are sometimes gaps in the train of thought, and exercises often require a significant conceptual leap from the preceding text. It was particularly annoying the way he suddenly switched to Polish notation for a while and then just as suddenly dropped it, without any obvious benefit. However, it is more accessible than most mathematical logic texts. The main competition for this text would be Ebbinghaus, which I prefer. The benefits of Enderton over that book are that it covers a wider range of topics and has a lot more exercises.

Terrific Book
Enderton's writing is the best I've seen in any introductory math textbook; he is lucid, well organised, comfortably paced but free of expository flab. The exercises (judging from chapters 2 and 3) are not terribly difficult, but quite useful in building one's intuition and connecting logic to other mathematics. I had the book for my Logic class as a first-semester sophomore with very little experience with proofs and no abstract algebra, and found it quite accessible. I guess the book starts off with an advantage, being about a subject as interesting as logic, but that does not seriously detract from its merit.

Still the best.
I review the classic FIRST EDITION. If you buy only one book on mathematical logic, get this one. It's by far the best logic book (see my other reviews) that is both 1)introductory and 2)sufficiently broad in scope and complete. The exposition is very clear and succinct- its suitable for beginners without getting wordy. Enderton always clearly explains what he's doing and why, keeping the reader focused on the big picture while going through the details. He helps to place topics in perspective, and has organized the book so readers can skip some of the more involved proofs and sections on the first reading.

Besides being easy to learn from, it's also the most rigorous introductory book I've seen-a rare combination. The proofs are detailed and complete, instead of the usual hand-waving or leaving everything as an exercise for the reader. There are some weak points in it, but overall you're not going to find a better book. It requires a little more 'mathematical sophistication' than most intro books- but if you've had some logic in a computer science course, or a little combinatorics or abstract algebra you'll be more than ready. Familiarity with automata/computability theory will help you in a few of the sections. Although Enderton is very good, it always helps to get several books on a subject- I'd recommend you pick up cheap copies of Boolos & Jeffrey's _Computability and Logic_ and Smullyan's _First-order logic_ as supplements.

Here is the complete table of contents for the first edition, c1972:

Chapter Zero - USEFUL FACTS ABOUT SETS. . . .1
Chapter One - SENTENTIAL LOGIC/ Informal Remarks on Formal Languages 14 /The Language of Sentential Logic 17/ Induction and Recursion 22/ Truth Assignments 30/ Unique Readability 39/ Sentential Connectives 44/ Switching Circuits 53/ Compactness and Effectiveness 58

Chapter Two - FIRST-ORDER LOGIC/ Preliminary Remarks 65/ First-Order Languages 67/ Truth and Models 79/ Unique Readability 97/ A Deductive Calculus 101/ Soundness and Completeness Theorems 124/ Models of Theories 140/ Interpretations between Theories 154/ Nonstandard Analysis 164

Chapter Three - UNDECIDABILITY/ Number Theory 174/ Natural Numbers with Successor 178/ Other Reducts of Number Theory 184/ A Subtheory of Number Theory 193/ Arithmetization of Syntax 217/ Incompleteness and Undecidability 227/ Applications to Set Theory 239/ Representing Exponentiation 245/ Recursive Functions 251

Chapter Four - SECOND-ORDER LOGIC/ Second-Order Languages 268/ Skolem Functions 274/ Many-Sorted Logic 277/ General Structures 281
Index 291 ... Read more

Book Description

Customer Reviews (1)

Good coverage, but lack of solutions weakens it.
Logic is one of the foundations of mathematics, making quality textbooks essential. This is one of the better ones, with descriptions followed by a large number of exercises. The basic strategy is to present the material in small sections, most of which are two pages or less in length. One unusual aspect of the book is that the authors chose to wait until chapter four to present truth tables. Traditionally the first topic in logic textbooks, leaving truth tables until later forces the reader to learn the operational meanings of the connectives. I approve of this pedagogical technique, undue reliance on truth tables can lead to the masking of some of the concepts of logic. Propositions and predicates are covered, with more ink spent on propositions that predicates and a simple set of axioms for addition is also presented
One negative point is that no solutions to the exercises are included. In my opinion, any book without solutions to some of the exercises is of reduced value as a textbook. Students work problems on their own and it is very important for them to get immediate feedback. This also reduces its' value as a text for self study.
I can recommend this book, the quality of the explanations and the number and detail of the exercises make that easy. However, the lack of solutions means that it can only receive my second highest recommendation.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Book Description

Customer Reviews (2)

Excellent introduction
I have to agree with the more recent reviewer and disagree with the first one.I don't even have college-level maths; in fact, I failed abysmally in my school-leaving maths exams (I think I got an F).I wanted to read this because, now in my mid-30s, I had got very interested in various mathematical topics (game theory, number theory, logic) and was sick of just reading popular scientific books about them that assumed that you didn't know how to read the symbols.I ordered this to get me started on logic.

Kleene does an excellent job of introducing a novice like me to the first principles; it's true that he doesn't hang about, and he has a way of bullying his readers into making the effort to understand by dropping sarcastic little remarks like 'Anyone who cannot follow this is clearly mentally sluggish', years of teaching logic in Madison, WI clearly finding payback right there.Some readers may find that kind of thing overbearing, but I found it bracing.I admit that I'm only on page 14, but already I can find the scope of a propositional connective, and when I woke up this morning I had never heard of such a thing.

I thoroughly recommend this book; a brisk, clear, ruthlessly no-nonsense introduction to the subject.Maybe it's not 'Mathematical Logic for Dummies', but Kleene would probably crack that dummies shouldn't be attempting the subject in the first place.

Not for the autodidact
Ten years ago, I took an undergraduate course in symbolic logic.Wishing recently to refresh my (extremely rusty) memories of the propositional calculus and the first-order predicate calculus, I picked up this meaty text and was extremely dismayed to find myself soundly defeated within the first few pages.Kleene does not even make a pretense of holding the reader's hand:either you get it or you don't.There is nothing even remotely "user-friendly" about this book's presentation of its material.

If one were to read this book under the guidance of a teacher, I think it might be worthwhile.It may not be fair for me to blame the author for my inability to understand his writing.If you're smarter than I am, you might breeze right through it.

I cannot recommend this book, though, good though it may be, for anyone who wishes to teach him/herself logic, nor for anyone who wishes to brush up on the subject.There are exercises for the reader to test his/her understanding of the material, but no answer key is provided.This is heavy-duty stuff, and not well-suited to the self-teacher. ... Read more

Book Description
This user-friendly introduction to the key concepts of mathematical logic focuses on concepts that are used by mathematicians in every branch of the subject. Using an assessible, conversational style, it approaches the subject mathematically (with precise statements of theorems and correct proofs), exposing readers to the strength and power of mathematics, as well as its limitations, as they work through challenging and technical results. KEY TOPICS: Structures and Languages. Deductions. Comnpleteness and Compactness. Incompleteness--Groundwork. The Incompleteness Theorems. Set Theory. : For readers in mathematics or related fields who want to learn about the key concepts and main results of mathematical logic that are central to the understanding of mathematics as a whole. ... Read more

Customer Reviews (1)

Most Accessible Undergraduate Text Covering Incompleteness
I have used this text in both graduate and undergraduate courses as well as tutorials and independent studies.It is the best text for a one semester course that introduces formal logic and has as its goal the Incompleteness Theorems of Godel.Students have reported it to be very readable and the array of exercises is excellent.Moreover, the author is a really nice fellow. ... Read more

Amazon.com
Mathematician John Allen Paulos bravely bridges the scientific and literary cultures with this amusing, enlightening look at numbers and stories. If you think those two things go together like a "horse and a paperclip," as Allen wryly observes, you only have to look at phenomena like the Bible codes, the stock market's ups and downs, and the Clinton sex scandal to begin to understand the hidden bonds between them. Put simply, mathematics can describe everything that happens, and everything that happens contextualizes mathematics. In demonstrating this, Paulos continues the noble numeracy crusade he began with A Mathematician Reads the Newspaper and Innumeracy. Perhaps the most compelling thought experiments in the book are those of the statistics of stereotyping and race relations. Paulos shows, mathematically, that minority status makes achieving equality extraordinarily difficult.

If you want to keep hold of your comfortable worldview, don't read Once Upon a Number. But you'll be missing out on an unforgettable reminder of what chance, coincidence, and odds really mean, along with several valuable life lessons that may help you understand lost socks, racism, and mistaken identity. --Therese Littleton Book Description
Named one of the Best Nonfiction Books of 1998 by the Los Angeles Times, Once Upon a Number presents a surprising and humorous look at the similarities between narratives and numbers.

"Both delightful and wise, this little book cries out to be kept close at hand, to be looked into from time to time, to be treasured as an old friend."-Los Angeles Times

A "charming narrative.…Almost every piece is fascinating."-Salon

"[Once Upon a Number] deserves rereading."-Booklist

Once Upon a Number shows that stories and numbers aren't as different as you might imagine. In fact, they have surprising and fascinating connections. Beside lucid accounts of cutting-edge information theory we get hilarious anecdotes and jokes; instructions for running a truly impressive pyramid scam; a freewheeling conversation between Groucho Marx and Bertrand Russell; explanations of why the mundane facts of the O.J. Simpson case are overwhelmingly incriminating; how the Unabomber's thinking shows signs of mathematical training; why we're much more likely to feel aggrieved than aggrieving; and dozens of other treats. America's most engaging mathematician has done it again. ... Read more

Customer Reviews (17)

A Mathematical Faery Tale
I've always marveled at the imagination of authors who write fiction. Having only really been good in Science and Mathematics in school, the literary world seemed incredible. I couldn't believe that someone would pick up a book and read about fictional characters for hours on end (or, if you were dyslexic like me - weeks on end!) It simply seemed quite unfathomable. Why would anyone do that?

Then at around 15, I got hooked (like many of my age and background) on Science Fiction e.g. Dune (Dune Chronicles, Book 1) Much later on, I even got into Fantasy e.g. Lord of the Rings J.R.R. Tolkien Boxed Set (The Hobbit and The Lord of the Rings) But I've never really got around to reading pulp fiction or romance novels - on any level. Spy novels or the occasional Agatha Christy murder mystery was as far as I dared travel down that road.

The problem was that I could never identify with the characters in those fictional stories. People with a scientific or mathematical bent, never seem to quite blend in to that world. In our worldview, the paradigms are much too different.

It is interesting to note that in the first few pages of the introduction, John Allen Paulos actually tries to parody characters that are found in such novels, augmenting them with some degree of Numeracy. It is easy to see, after just reading a few lines, why fiction remains a world where numbers are strictly forbidden!

This book is really a collection of essays. Each is entirely readable by itself. Unlike a novel however, it lacks the literary glue that compels us to keep reading it until the very end. And not all of the essays form a single unified focus, which is essential in presenting a profoundly new idea. It reminds me, of conversations I have with some of my students over a cup of tea in the evening: delightfully interesting but without some solid conclusion reached on any matter. Perhaps, this is the atmosphere that the author is trying to achieve as he talks about things as varied as the Bible Code to O. J. Simpson. In the last chapter he tries to bridge the gap between stories and statistics. However, by that time most readers would probably have come to the conclusion that the bigger the chasm between them - the better. I for one, would never want to read a historically correct version of Lord of the Rings, or a scientifically viable version of Dune.

Einstein was quoted as saying that. "Imagination is more important than knowledge."

Restraining our imagination to the "real" world would be just as much of a travesty as abandoning reality for myths and fantasy altogether. The important thing is in knowing how to tell the difference. And perhaps, this is what J. A. Paulos is really trying to bring across. In this day and age, you can't really take ANYTHING for granted, unless you first see the NUMBERS for yourself!

And this applies to the conclusions reached in THIS book too!

For the previous reviewer...
Usage Note: Assure, ensure, and insure all mean "to make secure or certain." Only assure is used with reference to a person in the sense of "to set the mind at rest": assured the leader of his loyalty. Ensure and insure are generally interchangeable.

Source: The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2000 by Houghton Mifflin Company.

The split that wasn't there
Paulos starts the book with a clearly absurd story, one that has numbers and statistics mixed in with a narrative. His question is, why is this so jarring? Why is it that people are literate and numerate, but so seldom both at once?

This book addresses that question. In part, he says that literature deals with many aspects of a few individuals, but statisticians generally study a very few aspects of very many individuals. He also notes that literature tends to treat each individual as a unique product of a unique time and place. In contrast, math and physics typically deal with cases where the specific individual is irrelevant. Any experiment on an electron gets the same result no matter which electron you use, or where, or when.

The dichotomy may not, in fact, exist. One could refer to the recent statistical studies of DNA data that have that literary much-about-few character and that often seek out the uniqueness of the study's subject. Part of Paulos's main point, however, is that reasoning in "human" prose is often just mathematical reasoning in street clothes. Other times, when day to day logic seems irrational to a simplistic "scientific" analysis, it turns out that there is a deeper kind of reasoning at work, and one that can be cast in formal terms.

Paulos delivers more than his nominal argument, though. His presentation is filled with little asides and self-referential humor. He is a logician after all, and, like the logician Lewis Carroll, uses his logic to create delightful unreason. Taken as a whole, it's a brief, enjoyable, and instructive look a the formal side of casual reasoning, and at the human side of mathematical logic.

//wiredweird

read "innumeracy" instead
I found this book disappointing. While some of the examples and anecdotes are interesting, and everything is very well written, I didn't really understand what the author's point actually was. I suggest that your time is better spent reading his other book, "Innumeracy", which is staggeringly good.

J.A. Paulos: a great mind.
The author deals in an original way with the difficult nexus between statistics and stories, between alpha- and betascience, without favouring one of them, and indeed arguing that both are complementary. This striking impartiality creates the space for original ideas about everyday-situations that every reader will certainly recognize. To be original about (seeming) banality, that is the work of a true great mind. The reason why I do not rate this book with the maximum score, is that it sometimes misses an overarching line of reasoning. The discursive path may strike some readers (like me) as too associative. Needless to say that such a style has its charms, but perhaps due to my Europe-continental education, I am a bit more at ease with a clear thesis and a transparant construction. Nevertheless: it is absolutely imperative to buy this book! ... Read more