Product Description

Focusing on the formal development of mathematics, this book shows readers how to read, understand, write, and construct mathematical proofs.Uses elementary number theory and congruence arithmetic throughout. Focuses on writing in mathematics. Reviews prior mathematical work with “Preview Activities” at the start of each section. Includes “Activities” throughout that relate to the material contained in each section. Focuses on Congruence Notation and Elementary Number Theorythroughout. For professionals in the sciences or engineering who need to brush up on their advanced mathematics skills.

Customer Reviews (3)

Terrible book on proofs.Avoid.
I can't even begin to illustrate my frustration with this book.With several colleges backing Dr. Sundstrom's work, I find it sadistically humorous that there are so many topographic errors and scatterbrained thoughts contained within it.The only thing I can postulate about colleges that chose this book is either a) they're getting money back for endorsing it, or b) there was a mathematical conference that overinflated this book.

Each chapter starts with throwing the reader into the deep end without explaining the necessary details.I can't get over the fact that this book was patronizing its reader in every chapter.I understand the concept of exemplary demonstration, but the examples are placed at the forefront of every section, rather than in the middle, meaning the reader has to sit back and guess for each of the problem, or jump a few pages ahead just to get past the opening section.This is fine for a reference manual, but this is unacceptable for a textbook designed for learning.

Problems frequently host topographical errors which tells me that no one proofread the print before sending it off for publication.Sure, an error here and there is something that every publisher has to live with, but this book has at least three errors in every section.If you're with an instructor that knows the material, this is doable, but the real problem is that this is, at core, a mathematics book.A simple typo means the entire problem changes, often times making the problem impossible to solve.

All that said, Dr. Sundstrom does a nice job explaining the particular elements of proofs, but some of the problems written are downright cruel to the student, giving problems that transcend the book's target audience.Still, most of those elements are defined in a confusing, but understandable way.

Also, it should be said that while there are answers to select problems in the back of the book, most of these are hints, and nearly none are completed proofs.If you're looking to learn by example, don't expect the exercise problems to help you.


All in all, I have to say that this is one of the worst math text books I've seen.Both under and overscoped, condescending, and full of errors; I can only warn those that read this with a big fat BUYERS BEWARE.

I expected better
I expected better from a college textbook with so many editors and reviewers, my classmates and I found errors in pretty much every lesson, typos that made it near impossible to get the answers are professor wanted turned in, and we did less than half the problems in the half of the book we went through!It was an easy to understand book, I probably could have taught myself the subject with this book, if it weren't for all the errors...also, a lot of questions were on the bottom of the right hand side, and the exercises were at the top of the other side, not exactly the easiest, especially when you're trying to share books.

Very Helpful
This book was very helpful.I have a better understanding of how to construct a proof and what proofs are best in each situation. The book also gave plenty of examples of proofs of theorems so you had something to compare your proof too.It also give some answers and hints to selected problems in the back which were a great help. ... Read more

Product Description
Chapters: Arithmetical Hierarchy. Source: Wikipedia. Pages: 42. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene hierarchy classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski-Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted and for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain set parameters. If a formula is logically equivalent to a formula with only bounded quantifiers then is assigned the classifications and . The classifications and are defined inductively for every natural number n using the following rules: In other words, a formula is equivalent to a formula that begins with some existential quantifiers and alternates times between series of existential and universal quantifiers; while a formula is equivalent to a formula that begins with some universal quantifiers and alternates similarly. Because every formula is equivalent to a formula in prenex normal form, every formula with no set quantifiers is assigned at least one classification. Because meaningless quantifiers can be added to any formula, once a formula is a...More: http://booksllc.net/?id=186475 ... Read more

Product Description
Chapters: Primitive Notion, Truth Value, Lcf, Infinite Set, Shelah Cardinal, Singular Cardinals Hypothesis, Higher-Order Logic, Bunched Logic, Reinhardt Cardinal, Extensionality, Institutional Model Theory, Kleene-rosser Paradox, Rank-Into-Rank, Kunen's Inconsistency Theorem, Double Negation, Substructural Logic, Unfoldable Cardinal, Indescribable Cardinal, Judgment, Shrewd Cardinal, Impossibility of a Gambling System, Ω-Logic, Truth Table Reduction, Zero Dagger, Clone, Coherent Space, Complete Theory, Hartogs Number, Ordinal Definable Set, Strongly Compact Cardinal, Jónsson Cardinal, Erdős Cardinal, Martin's Maximum, Ramsey Cardinal, Tav, Subtle Cardinal, Richardson's Theorem, Subcompact Cardinal, Supernatural Numbers, Logical Graph, Universally Baire Set, Streett Automaton, Disjunction and Existence Properties, Rabin Automaton, Sudan Function, Mu Calculus, Pseudo-Order, Generic Filter, Classical Mathematics, Krull's Theorem, Trichotomy, Kuratowski's Free Set Theorem, Axiom Schema of Predicative Separation, Least Fixed Point, Borel Equivalence Relation, Rowbottom Cardinal, Code, Ineffable Cardinal, Additively Indecomposable Ordinal, Internal Set, Homogeneous Tree, Categorical Set Theory, Vopěnka's Principle, Admissible Ordinal, Conservativity Theorem, Hereditarily Countable Set, Heyting Arithmetic, Cabal, Many-Sorted Logic, Ramified Forcing, Double Recursion, Hereditary Set, Finite Character, Superposition Calculus, Semicomputable Function, Axiom of Limitation of Size, Paraconsistent Mathematics, Suslin Representation, Robinson's Joint Consistency Theorem, Recursive Ordinal, Critical Point, Reduced Product, Normal Form, Takeuti Conjecture, Minimal Logic, Kleene-brouwer Order, Θ, Axiom of Projective Determinacy, Star-Free Language, Extendible Cardinal, Reflecting Cardinal, Cantor-Dedekind Axiom, Equinumerosity, Laver Function, Herbrand Interpretation, Kleene Fixed-Point Theorem, Lindström's Theorem, Material Nonimplication,...More: http://booksllc.net/?id=4623947 ... Read more

Product Description

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability.

The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory.

Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises.

Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.

Customer Reviews (1)

used early draft as grad text
I took a great graduate course from Prof. Andrews, way back in the 1970's, where his class lecture notes were titled "To Truth Through Proof", so I assume that was a very very early draft of this book.

If so, this must be a very good book, because his notes were wonderful even back then. ... Read more

Product Description
Philosophical considerations, which are often ignored or treated casually, are given careful consideration in this introduction. Thomas Forster places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in an original analysis of well established topics.The presentation illustrates difficult points and includes many exercises. Little previous knowledge of logic is required and only a knowledge of standard undergraduate mathematics is assumed. ... Read more

Customer Reviews (1)

Unusual reliance on text rather than equations
As a regular reviewer who reads the books he reviews, I am beginning to find a routine sameness to many of the books I examine. This holds for introductory logic, some areas of which have not changed for thousands of years. In many cases, it has become so routine that it is very hard to read the books, and to solve this problem I have adopted the solution that I read approximately ten pages at a time. Most books covering this material are a sequential listing and proving of the fundamental theorems. Correct, but hardly a point of differentiation between books.
Fortunately, I found this book to be different. The author uses more explanatory text and fewer equations than most others do. While that alone does not make it better, the quality of the exposition does. There was more enthusiasm generated when reading this book than I have had in a long time when reading an introductory mathematics book. There was no reason to adopt the ten page rule. Despite the relative lack of equations, all of the basics are covered in sufficient detail. One hundred and thirty eight exercises are interspersed within the text and solutions to approximately one fourth of the exercises are included.
A book that can be used as a textbook or for self-study, it manages to be different enough to read easily, but not so different that it is difficult to learn from. ... Read more

Product Description
This completely self-contained study, widely considered the best book in the field, is intended to serve both as an introduction to quantification theory and as an exposition of new results and techniques in "analytic" or "cut-free" methods. Presented in tableau format, the material covers propositional and first-order logic. 144 illustrations. ... Read more

Customer Reviews (8)

Great stuff.
First, this isn't one of Smullyan's popular puzzle books- its a serious mathematics text. Second, don't use this as your first exposure to first-order logic (note the title doesnt say "Introduction to ...")- although logically self-contained, it requires some experience to appreciate what a neat little book this is.

It's not a general mathematical logic text- there is no model theory (beyond basic Skolem-Lowenheim), incompleteness, recursion theory, or set theory. It covers tableaux (this alone is worth the price of the book), Hilbert-style axiomatic systems (briefly), sequent systems, Gentzen's Hauptsatz and Extended Hauptsatz, Craig's and Beth's theorems, and more. But the heart of the book is completeness theorems, their proofs, and closely related material such as compactness and Herbrand-like theorems. Smullyan shows there are two main approaches to completeness (analytic vs. synthetic), breaks each into stages, provides nice abstracted formulations, and usually gives several different proofs of each result. The centerpiece is his "Fundamental Theorem of Quantification Theory", a theorem associating a truth-table tautology with every valid first-order sentence (check out the amazingly slick proof of completeness for the the Hilbert-style system that this provides). Similar constructions such as magic sets are also discussed. All this forms a much more extensive and illuminating look at completeness proofs than I've seen elsewhere.

The first-order logic used in the book has no equality and no function signs. There are few exercises, most of them simple. Smullyan writes clearly and with an appropriate amount of rigor (but its not as polished as his later books). Makes a great supplement to more general-purpose introductory mathematical logic books. If you haven't seen the tableau method yet buy this book immediately. Experienced readers will appreciate the sophisticated coverage of completeness proofs.

An Oddity But a Good-ity. Wait, that's terrible.
The reviewer from Illinois gave a very good characterization of Smullyan's style here:
"Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. No discussions, ala WvO Quine, on the history or linguistic difficulties of a concept, just definition and proof."
Readers familiar with Smullyan's enormous talent for popular exposition may be expecting the same herein: not so. This is very much for people who have attained what medical professionals call "mathematical maturity" (which is about as difficult to attain as zen, yet perhaps amounts to little more than the ability to read VCR instruction manuals). For example, the very first section is a wiz-bang treatment of trees (not the usual graph-theoretic ones), defined in the abstract/axiomatic fashion.
Of course, people who spend perhaps way too much of their time steeped in math are attracted to treatments of just this sort.
A structural characterization in terms of sets and mappings is much more meaningful, interesting, and aesthetically pleasing to those with these unusual inclinations (compulsions?) than a characterization framed significantly by historical motivation (please understand that I'm speaking roughly here). This is why I gave a positive review. A star was witheld for the selfish reason that I'm not sure I'll find much use for such an odd treatment of model theory, the topic for which I was seeking a more mainstream treatment when I purchased this. Regrets are nonetheless few: time spent reading Smullyan is never a waste.

a classic
I mainly bought this book because of the influence it has had on numerous modern-day logic texts. If you are unfamiliar with the tableaux method for structural proofs, then you will gain alot from reading this, as it provides a different perspective from the more popular Hilbert-system approach. Tableaux systems, of course, have been made popular because they are easy to program with a computer. Please see Gallier's "Logic for Computer Scientists" for more on this matter.

Great as a Reference, Probably Not for True Beginners
This is an excellent reference!It has more material covered in just 155 pages than most other works address in twice as much space.I refer to it very often.

However, I doubt it would be appropriate for someone that has not previously been introduced to the material.If a truly introductory text is required, I would look elsewhere.

A Stripped-Down Exposition of a Bare-bones Subject
This is a book by a man I knew for his books of puzzles-chatty books of great originality that have fun with the paradoxical possibilities of logic. Here he is the teacher of logic, and aside from an occasional phrase, the serious mathematician. However, Smullyan's originality shines through in this book as well. He presents logic as a branch of mathematics rather than an abstraction of ordinary language. And he uses a method from the recent literature, tableaux, to build his proofs in a simple and satisfying way. He gets directly to the main result as to the provability of valid sentences using this method for both the propositional calculus and the predicate calculus.

Smullyan procedes rapidly because he makes some assumptions about the reader's knowledge. The reader must understand the difference between mathematics and meta-mathematics-that is, should be able to separate out the talking about the sentences of the system, which may contain (among other signs) the conjunction, disjunction, and negation, from the more-or-less informal arguments that prove assertions about these sentences using natural language, with its "and", "or", and "not". Moreover, the concept of "proof" is used at two levels: the particular tableau that constitutes a proof of a sentence, and the "proofs" about tableaux and other concepts of the "system".

Besides this, the reader should have a good feel for recursive definitions, which are used everywhere. Finally, this model reader should know the difference between countably-infinite sets and uncountably-infinite sets.

I knew all that, but still found the text slow going, maybe because I have been away from mathematics for decades. But there is another reason, too. Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. No discussions, ala WvO Quine, on the history or linguistic difficulties of a concept, just definition and proof. This is an abstraction of a subject which is already an abstraction. So I usually found myself trying to understand what it all meant, in other than these stark set-and-mapping terms. On the other hand, many difficulties caused by the details of historical development of the subject vanish, and the results stand-... simple, directly derived.

This is a slender Dover volume, of high quality and low cost. I would have given the book 5 stars, but for two things. The exercises are too hard, sometimes, and without answers, and the index is very poor. Still, I think the treatment is the best around for those who want to use logic as a basis for studying incompleteness or proof theory. It is not to be confused with a more full-blown treatment that also treats logic as a branch of the humanities. ... Read more

Product Description
Graduate-level historical study is ideal for students intending to specialize in the topic, as well as those who only need a general treatment. Part I discusses traditional and symbolic logic. Part II explores the foundations of mathematics, emphasizing Hilbert’s metamathematics. Part III focuses on the philosophy of mathematics. Each chapter has extensive supplementary notes; a detailed appendix charts modern developments.
... Read more

Customer Reviews (1)

Nothing Great.
This is a recently issued Dover reprint of a 1963 Van Nostrand book. It contains a rough coverage of some mathematics, some history, and some philosophy. You need some familiarity with mathematical logic to read the book, but it certainly isnt "graduate-level" as claimed in the editorial review. It's divided into three parts. Part 1 is mostly a shoddy and outdated introduction to symbolic logic. People who dont already know this material will probably be confused by the presentation, and people who have taken a class in mathematical logic will find it dull and clumsy - skim it. Part 2 is a history of the major developments in mathematical logic and foundations from around 1870 to 1940. This coverage is ok but dated, not so much because of the omission of more recent events, but for the lack of critical perspective from the period Kneebone is discussing. He reviews all the historical material you would expect in an introductory survey, without offering anything particularly insightful, and sometimes fails to emphasize the truly important innovations against the background of lesser technical apparatus. Part 3 is a lightweight look at philosophy of mathematics, the longest section focusing on Whitehead's epistemology.Overall the book is routine, broad but not deep, and somewhat outdated. ... Read more

Product Description
This volume gives an overview of linear logic in five parts: category theory; complexity and expressivity; proof theory; proof nets; and the geometry of interaction. The book includes a general introduction to linear logic that will ensure this book's use by the novice as well as the expert.Mathematicians and computer scientists will learn much from this book. ... Read more

Product Description
Logic forms the basis of mathematics and is a fundamental part of any mathematics course. This book provides students with a clear and accessible introduction to this important subject, using the concept of model as the main focus and covering a wide area of logic. The chapters of the book cover propositional calculus, boolean algebras, predicate calculus and completelness theorems with answeres to all of the exercises and the end of the volume. This is an ideal introduction to mathematics and logic for the advanced undergraduate student. ... Read more

Customer Reviews (1)

OK but Hard
You'll find this very hard unless you are a competent math major at one of the better universities. Similar to Elliot Mendelson's text, but not quite as good. Good chapter on Boolean algebra as a
piece of pure math; Halmos and Givant is gentler, though.

Interesting topic covered: the resolution so dear to the AI crowd. Unlike most mathematicians, Cori and Lascar have time for
the way computer scientists think. At the same time, this book does not cover tableau methods (see Smullyan), natural deduction, Genzen's ideas, and so on. For pure logic at the advanced undergrad level, you're better off with Bostock.

Haven't seen Part II, so cannot comment on the treatment of set theory. This is something Mendelson and Machover already do well. ... Read more

Product Description
Elementary set theory accustoms the students to mathematical abstraction, includes the standard constructions of relations, functions, and orderings, and leads to a discussion of the various orders of infinity. The material on logic covers not only the standard statement logic and first-order predicate logic but includes an introduction to formal systems, axiomatization, and model theory. The section on algebra is presented with an emphasis on lattices as well as Boolean and Heyting algebras. Background for recent research in natural language semantics includes sections on lambda-abstraction and generalized quantifiers. Chapters on automata theory and formal languages contain a discussion of languages between context-free and context-sensitive and form the background for much current work in syntactic theory and computational linguistics. The many exercises not only reinforce basic skills but offer an entry to linguistic applications of mathematical concepts. For upper-level undergraduate students and graduate students in theoretical linguistics, computer-science students with interests in computational linguistics, logic programming and artificial intelligence, mathematicians and logicians with interests in linguistics and the semantics of natural language. ... Read more

Customer Reviews (3)

Extremely good.
This book is incredible. It's a great introduction to axiomatic set theory at the beginning, and it does a great introduction to model theory. I haven't gotten to the linguistics part of it yet, but it covers math and logic in an extremely effective way.

Comprehensive
This book is well written and detailed.I found it particularly useful for my semantics course.It covers the necessary logic one need to do semantics.It also discuses type theory and the lambda calculus. This book is a great complement to any semantics text.

Not Just for Linguists
Mathematical Methods in Linquistics is far more about mathematical methods than about linguistics, although in many places linquistics is used as a source of examples.

Instead it covers such mathematical topics as sets (including infinite sets), relations, a good deal of mathematical logic,
automata (up to turing machines), the lambda calculus, lattices and more.

This would be an excellent book for an advanced undergraduate or graduate student in either mathematics or computer science to use
either as a review text, or as a study guide for further investigation. ... Read more

Product Description
Stimulating, thought-provoking analysis of a number of the most interesting intellectual inconsistencies in mathematics, physics and language. Delightful elucidations of methods for misunderstanding the real world of experiment (Aristotle’s Circle paradox), being led astray by algebra (De Morgan’s paradox) and other mind-benders. Some high school algebra and geometry is assumed; any other math needed is developed in text. Reprint of 1982 ed.
... Read more

Customer Reviews (8)

An adequate introduction
If you've never encountered the problems in MFaP before, you're sure to find MFaP a decent and generally easygoing introduction to the subject matter.If you have encountered them before, you're sure to find little new between the covers of this slim volume.MFaP is an able and by-the-numbers overview of an exceedingly complex and fascinating topic.Should you read to the endof it, I'd highly recommend having a look at the brief bibliography Bunch assembled.Not all of it represents 2008 state-of-the-art, but there are several outstanding titles on this list to whet the appetite for further study.Read MFaP to taste some choice bits.But be sure to read elsewhere if you it's a feast you're after.

A great introduction to the limits of math
Most causual users of math consider it to be the most unassailable of endeavors.

After, 2 + 2 always has to equal 4 doesn't it?

It turns out that at the periphery of math there are certain inconsistencies that can arise either owing to the use of faulty methods in arriving at a conclusion (what Bunch calls "fallacies") or inconsistencies owing to the limits of math itself (what Bunch calls "paradoxes").

Though one would need recourse to the book itself in order to completely understand what Bunch means by each category, what follows are a couple of examples to help illustrate the kinds of issues this book will treat.

In relation to fallacies, an early example used by Bunch is Aristotle's paradox wherein Aristotle tried to use a deceptively simple experiment to measure the perimeter of two circles.For ease of convenience, let's say he used two coins of different denominations...say a dime and a half dollar.

Obviously, the coins by their size have to have different measures of distance around their perimeters.And yet, according to Aristotle's experiment, they turn out equally.They turn out equally because Aristotle simply placed one on top of the other and rolled them to see which would make a complete turn the earliest.As you may have gleaned they both turned at the same time owing to the particular mathematics of circles.

Bunch's point is that by applying incorrect reasoning Aristotle's "paradoxical" result was simply a fallacy.

In terms of true paradoxes, Bunch discussed Kurt Godel's incompleteness theorem which says that any consistent system will produce so called "formally undecideable propositions."In other words, to the extent that a consistent system produces self referential statements, those statements can defy formal proof.

An oft used English language example is "This sentence is false."Obviously, the sentence is neither be bracketed with all true statements or all false statements owing to its category defying nature.

In turns out that Kurt Godel was able to stand over two millenia of math philosophy on its head by showing that math had its logically limits of proof.

As can be seen from the previous examples, this book is thought provoking even for casual readers who admittedly will have to struggle cracking the hard nutshell of some its more dense arguments.However, those who do so will be richly rewarded for the heightened understanding of the limits of math they have thereby gained in the process.

Informal and engaging
This is a great informal treatment of some of the more notable paradoxes and fallacies of mathematics and mathematical reasoning, old and new. Bunch's prose style is clear and unencumbered and his presentation of each topic - from his easily resolved fallacies and paradoxes of basic algebra and geometry to the deeper and unresolved paradoxes of logic and analysis - is always clean, well-illustrated and engaging.

At a glance, he treats:
The Liar paradox and Godel's Incompleteness theorems
Zeno's and the Sorites paradoxes and the conceptual difficulties associated with the continuum
The existence of irrational magnitudes and some basic philosophical issues associated with existence proofs
The Petersburg paradox
The paradoxes of Infinity and the Formalist and Intuitionist responses to them
The set theoretic paradoxes of Cantor, Russell, and Burali-Forti
The paradoxes of the axiom of choice including the Cantor diagonilisation, Skolem, Hausdorff and Tarski-Banach parodoxes

and a range of thought experiments which highlight the difficulties that may be asociated with applying abstract reasoning to the real world - notably those of the Thompson lamp experiment and Tarski-Banach golden sphere manufacturing plant.

If you want a good popular treatment of the subject matter with a detailed and informal emphasis on the key themes mathematical logic, then this is the book for you. The informal description Godel's first Incompleteness theorem is excellent, as is the discussion of the paradoxes of self reference as they appear in set theory and logic. As such, I would recommend it as excellent recreational reading for anyone with a budding interest in mathematical logic, whether they be math graduates or high-school students.

This is a Great Book for Math Fans
This is a great book for people who love mathematics, including: recreational math enthusiasts, math teachers, professors and other university level math instructors, curious and self-motivated students, etc. This book provides numerous examples of how seemingly logical steps can lead to mathematically fallacious results. The level of math ranges from advanced high school to college level math, but the level is not really important ... what is important is the insights one can get from looking at common mathematical mistakes.

This book may also be of interest to neuroscientists, cognitive scientists, and psychologists who are interested in how human beings learn and apply mathematics. On a somewhat related note, I have noticed that (for some strange reason) this book has attracted a set of rather bizarre reviewers (see below). Please ignore them and buy this inexpensive and insightful book on math.

Zeno and set theory
It is the paradoxes that keep us honest in mathematics. Tarski with Banach found a basic flaw in the axiom of choice in set theory. Zeno has puzzled children for two thousand years... Time travel paradoxes are the modern "new" problem of tacyonic loops and the Hawking conjecture. Without examples of critical thinking doctrine rules and men become fools! ... Read more

Product Description
Most popular Russian puzzle book ever published. Marvelously varied puzzles ranging from simple "catch" riddles to difficult problems. Lavishly illustrated with clear diagrams and amusing sketches. Edited for English-readers, while retaining warmth and charm of original. Inexpensive edition of first English translation. Introduction by Martin Gardner. 425 line illustrations. Solutions.
Amazon.com Review
This book has been a classic in the former Soviet Union sinceit was first published in 1956, and it remains just as entertainingtoday. A master at making math fun for his high school students, BorisKordemsky loaded this clever collection with a wide variety of mathand logic related games and puzzles dealing with magic squares, trickyweights and measures, properties of numbers, mathematical tricks, andmore.Number and math game fans are bound to find several newamusements here. Even many of the well-known classics from generationspast take on new life with the fresh twists Kordemsky provides. ... Read more

Customer Reviews (13)

Jewel of math history!
This book, literally, is one of the best puzzle books ever published in history, capturing the world's greatest, most classic math problems and concepts!
The puzzles range from simple brain-busters to in-depth number theories, so there are puzzles for mathematicians at any age and level.

As usual, Gardner rocks
No surprise. If you knew Gardner in the "good times" when he wrote in Scientific American you will have no surprise. Very good book for enjoyment when in a warm afternoon in a hammock.

In Soviet Russia, the puzzle solves you
In Soviet Russia, the puzzles are also converted into practical problems involving steel working, railways and Young Pioneers (well, not all). But as with many older books (and also Soviet books), what they gain in a certain charm or obscurity, they lose in clarity or straightforwardness of presentation. So that you could make many problems at least 10-30% easier to decipher just be rewriting the descriptions.

I haven't read that many puzzle books yet, but this is probably my second favorite so far. My favorite is: Professor Stewart's Cabinet of Mathematical Curiosities.

Funny, challenging, and well written!
I bought this book while working as a gifted children teacher. I liked it so much that I used to keep it in my car and solve riddles whenever I had to wait for someone. It is a great resource for all teachers; children are suddenly made quiet when you present a puzzle to them.

I especially like the stick puzzles, where you can distribute a number of matches to students (by the way, it works with kids, teenagers and adults alike) and give them a puzzle. The advantage of this kind of puzzle is that you can give additional tasks to those fast-solvers; you do not have a story behind it.

The organization of the book is excellent; it is divided by difficulty levels as well as by type of puzzle. For example, you have different levels of geometry problems and of sticks problems.

Great book!

very good
despite of difficulty, I love it because there are various good problems
Thanks you ... Read more

Product Description
Venn, best known for his diagrams for set theory, primarily studied logic and probability theory. The present book is a study of the principles of logic, with special emphasis on inference and induction. From the Preface to the First Edition (1889): ``As many readers will probably perceive, the main original guiding influence with me--as with most of those of the middle generation, and especially with most of those who approached logic with previous mathematical or scientific training--was that of Mill ... I still continue to regard the general attitude towards phenomena, which Mill took up as a logician, to be the soundest and most useful for scientific study ... '' From the Preface to the Second Edition (1907): ``Though thus leaving the main outlines unaltered I have done what I could to improve the work, and to try to bring it up to date ... A number of paragraphs have been altered, others have been re-written, and many hundreds of minor alterations, additions and corrections inserted ... '' ... Read more

Product Description
Mathematical Proofs: A Transition to Advanced Mathematics, 2/e, prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets. KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory. MARKET: For all readers interested in advanced mathematics and logic.
... Read more

Customer Reviews (17)

Appropriate for beginning the trek into advanced math.
It's an all around great book. Some people might complain about it holding the reader's hand too much, but that's what makes the book so great. It assumes you never seen aproof before, and it helps guide you towards writing and analyzing proofs. The examples provided are clear. The section problems for the most part mirror the examples provided, so usually you have a model solution to work with. All in all, it's a great book for self study. If you have to purchase it for a class your instructor made an excellent choice.

A must read for the major!
If you plan to take any seconed year math courses such as linear algebra, calculus or real analysis I strongly suggest you read this book. I found that some seconed year books assume that you have some knowledge of how proofs are constructed. This book assumes that you know nothing about proofs.It cleared up a lot of confusion for me. I am pleased with book and it was well worth the price.

A very solid introduction to mathematical thinking
Being a physist and a computer scientist i find this book very helpful for someone to aquire good skills in the mathematical language and thought.
I consider that these two principles enable the creativity in mathematics for someone less involved with formal mathematical thought

Superb
Thisbook is an excellent introduction to the basic of mathematical logic and proofs, the latter laying the foundation for advances areas of mathematics such topology, algebra, analysis, among others.

Highly recommened in conjunction with Vellemen's wonderful text.

Dock 1 star for comparatively hefty price (in this regard, see Velleman and Eccles comparative cheapness!).

Gripping and Suspenseful
truly a mystery on every page, you never know the answer till the very end! ... Read more

Product Description
An accessible introduction to abstract mathematics with an emphasis on proof writing

Addressing the importance of constructing and understanding mathematical proofs, Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers introduces key concepts from logic and set theory as well as the fundamental definitions of algebra to prepare readers for further study in the field of mathematics. The author supplies a seamless, hands-on presentation of number systems, utilizing key elements of logic and set theory and encouraging readers to abide by the fundamental rule that you are not allowed to use any results that you have not proved yet.

The book begins with a focus on the elements of logic used in everyday mathematical language, exposing readers to standard proof methods and Russell's Paradox. Once this foundation is established, subsequent chapters explore more rigorous mathematical exposition that outlines the requisite elements of Zermelo-Fraenkel set theory and constructs the natural numbers and integers as well as rational, real, and complex numbers in a rigorous, yet accessible manner. Abstraction is introduced as a tool, and special focus is dedicated to concrete, accessible applications, such as public key encryption, that are made possible by abstract ideas. The book concludes with a self-contained proof of Abel's Theorem and an investigation of deeper set theory by introducing the Axiom of Choice, ordinal numbers, and cardinal numbers.

Throughout each chapter, proofs are written in much detail with explicit indications that emphasize the main ideas and techniques of proof writing. Exercises at varied levels of mathematical development allow readers to test their understanding of the material, and a related Web site features video presentations for each topic, which can be used along with the book or independently for self-study.

Classroom-tested to ensure a fluid and accessible presentation, Fundamentals of Mathematics is an excellent book for mathematics courses on proofs, logic, and set theory at the upper-undergraduate level as well as a supplement for transition courses that prepare students for the rigorous mathematical reasoning of advanced calculus, real analysis, and modern algebra. The book is also a suitable reference for professionals in all areas of mathematics education who are interested in mathematical proofs and the foundation upon which all mathematics is built.

Product Description
Stat Labs: Mathematical Statistics Through Applications blends mathematical statistics with modern statistical practice. It turns the traditional teaching of mathematical statistics on its head by making a case study the centerpiece of each chapter. Chapters begin with the introduction of a real problem followed by a description of the data collected to address the problem, rich background material to put the problem in context, and suggestions for investigating the problem. This novel approach to bringing data analysis into the theoretical course is ideal for motivating and illustrating standard statistical techniques, for helping students understand mathematical statistics, and for showing how statistics can be useful in a wide variety of contexts. Stat Labs is designed for use in a calculus-based introductory statistics course. It would be equally effective as a stand-alone text or as a supplement to a traditional mathematical statistics text. Deborah Nolan received her Ph.D in Statistics from Yale University, and she is now Professor of Statistics at the University of California, Berkeley. Her research interests include high-dimensional modeling, the teaching of statistics, and the use of technology in education. She has also been involved in encouraging women into research careers in the mathematical sciences, and in 1997 edited the volume, Women in Mathematics: Scaling the Heights for the Mathematical Association of America. Terry Speed's professional activities center around teaching and research involving th application of statistics to genetics and molecular biology. He spends 50% of his time in the Department of Statistics of the University of California at Berkeley, and the other 50% with the Genetics and Bioinformatics Group of the Walter & Eliza Hall Institute of Medical Research in Melbourne, Australia. ... Read more

Customer Reviews (2)

interesting approach to teaching statistics


I took my introductory mathematical statistics course from Brad Efron and Charles Stein at Stanford. It gave me a strong foundation and understanding of decision theory, hypothesis testing and the power of exponential families of distributions. Although I understood the Neyman - Pearson theory of hypothesis testing and confidence intervals, I was not completely comfortable applying what I had learn when confronted with real problems. Students even at the best universities in the US still need consulting experience before they are comfortable with the tools they learn.
That is why this book so intrigued me. It approaches the theory and methods from applications first. Each chapter poses an interesting real problem and then progresses to a solution, introducing only the necessary tools. Many important statistical tools and topics are covered this way. It is particularly good for students interested in biostatistics as many of the applications fall in that area. In fact the subject of maternal smoking and infant health is treated in Chapter 1 and revisited in Chapter 10 with additional data to consider.

It is designed for an advanced undergraduate course for statistics majors and has been successfully implemented by the authors at Berkeley. I think this can really work and make mathematical statistics interesting. However, the student should not think that he or she can come out of this class and jump right into consulting. The biggest problem in consulting is working with the client to pose a well formulated problems that address their questions. In this text the authors have already done that long and difficult task for us and we are left in a position to learn the subject matter and the statistical tools needed for the solution.

An Excellent Book
This book is now being used at the university I attend for a Data Analysis Course.The examples provided were used as projects for out class.I learned a lot about real world problems that can be solved and modeled using statistics.

I recommend the software MiniTAB to accompany this book.

I especially liked the chapter on HIV and mortality (very interesting results)! ... Read more

Product Description
Sweet Reason is an introductory text for courses on modern logic unlike any other.The basic rudiments of formal and informal logical are here, all clearly described. Further, it focuses students on the real world, where the discipline of logic adds substance and meaning to all kinds of human discourse.Everything from puzzles, paradoxes, and mathematical proofs to campaign debate excerpts, government regulations, and cartoons are used to show how logic is put to work by philosophers, mathematicians, advertisers, computer scientists, politicians, and others.As the book alternately discusses, instructs, questions, teases, and challenges, readers will find themselves: 1) absorbing the fundamentals of the discipline, 2) becoming fluent in thte language of logic, 3) understanding how logic works in the real world, 4) enjoying logic's ability to entertain, surprise, subvert, and enlighten. ... Read more

Customer Reviews (3)

Do not take a class that requires this book.
This book is horrible. It skips around, doesn't explain things well, and is definitely not a beginner's textbook for Logic. The authors included a lot of humor in the book, which helps make the struggle through each page more interesting, but there are much better textbooks for Logic, like The Logic Book.

the best logic book ever
If you want a boring book that tells you exactly what to do to get a good grade from your teacher, this is not it. If you want an interesting book that gets to the heart of modern logic with a rich collection of examples in real real life (as opposed to fake real life) ranging from politics to Donald Duck comics, that encourage you to think for yourself (sadly, no-one can make you think for yourself), that gives you a sense of why people care about the subject and why you should too --- this is that book. I cannot praise it highly enough.

logic textbook
If your class requires this book...drop it and take something else. The textbook is written to be clever and funny instead of instructional and is by far the worst textbook I have ever used. That being said, the seller shipped very fast and the book was in great condition. It's not their fault the book authors are clearly sadistic. ... Read more

Product Description
Volume II, on formal (ZFC) set theory, incorporates a self-contained "chapter 0" on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques provides a solid foundation in set theory and a thorough context for the presentation of advanced topics (such as absoluteness, relative consistency results, two expositions of Godel's construstive universe, numerous ways of viewing recursion and Cohen forcing). ... Read more

Customer Reviews (1)

Excellent book as an advanced introduction.
My opinion is that this is the most readable and user friendly advanced book on Set Theory today. Especially the treatment of (Cohen's) Forcing is valuable, since other expositions on this subject were not suitable for beginning graduate students (Kunnen - very good, but very sophisticated,Jech - also good, but presents Forcing via Boolean Algebra, which is less widely used today, and the book is too long as an advance introduction).
Also the style or writing is inviting, and not intimidating.
Very significant book, pedagogically. ... Read more

Product Description

This comprehensive introduction presents the fundamentals of symbolic logic clearly, systematically, and in a straightforward style accessible to readers. Each chapter, or unit, is divided into easily comprehended small “bites” that enable learners to master the material step-by-step, rather than being overwhelmed by masses of information covered too quickly. The book provides extremely detailed explanations of procedures and techniques, and was written in the conviction that anyone can thoroughly master its content. A four-part organization covers sentential logic, monadic predicate logic, relational predicate logic, and extra credit units that glimpse into alternative methods of logic and more advanced topics. For individuals interested in the formal study of logic.

Customer Reviews (11)

Oh Logic...
I ordered this book for a college Logic class, and it hasn't disappointed. There's no getting around the somewhat boring text in this book, even though it is flavored by some humorous logic scenarios. But, I fault my lack of interest in the subject, and not the book itself. I've learned a lot from the class, and the book is essential.

It arrived in great condition, even though I ordered it used, and it had a speedy delivery.

Excellent textbook
I really like this book. It's an excellent undergraduate introduction to logic for non-mathematicians. It's not dumbed down in any way. It's rigorous. The exercises are challenging and plentiful. Most importantly, it explains the basic concepts and tools of symbolic logic in a clear, unpretentious manner. (The 5th edition corrects the mistakes of the 4th.)

By comparison, Copi seems unfocused and flabby; Bergmann & Moor comes off as unnecessarily muscular given the level of the material.

An excellent choice.

Has its ups and downs...
This is a decent textbook. The third edition (this edition) is almost EXACTLY identical to the fourth edition, all the exercises are the same, the only difference being the page numbers. Despite being overly wordy in more places than one (see: Unit 1-26, i.e. the whole book...) some of the examples and exercises can be rather amusing. It may seem like an overwhelming amount of material at first, but if you stick with it, it really isn't that difficult. As long as you take it slowly, and your instructor is able to explain things in simpler terms than the book, you'll do just fine.

Excellent Start in Logical Studies
This book is the best introduction to logic on the market.Others have made this clear--incuding professors of logic (of both the philosophical and mathematical sort).The reason why I am writing this is because I was flabbergasted to read a couple of the reviews of this book that appear on this website. Anyone that rates this book under four stars was perhaps unsuccessful in logic, tried to blame their lack of ability on Klenk's wonderful text, and didn't have enough ability to begin logic anyway.Logic is difficult.However, this book is completely readable, assuming the one reading it is careful.People that want to be more spoon-fed than how this book attempts, should not begin the topic.

Better than most logic books out there!
After attempting an intro to symbolic logic course using the text, "The Logic Book", this other text "Understanding Symbolic Logic" was a godsend! It is very useful and easily understood. I really hope that professors require this as the primary text for their course. Perhaps, if this book is used rather than "The Logic Book", then maybe students will actually pass the course!!!! ... Read more