Book Description

Kurt Godel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computation theory, as well as for the strong individuality of his writings on the philosophy of mathematics. Less well-known is his discovery of unusual cosmological models for Einstein's equations, permitting "time-travel" into the past.

This second volume of a comprehensive edition of Godel's works collects together all his publications from 1938 to 1974. Together with Volume I (Publications 1929-1936), it makes available for the first time in a single source all of his previously published work. Continuing the format established in the earlier volume, the present text includes introductory notes that provide extensive explanatory and historical commentary on each of the papers, a facing English translation of the one German original, and a complete bibliography. Succeeding volumes are to contain unpublished manuscripts, lectures, correspondence, and extracts from the notebooks.

Collected Works is designed to be accessible and useful to as wide an audience as possible without sacrificing scientific or historical accuracy. The only complete edition available in English, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science. These volumes will also interest scientists and all others who wish to be acquainted with one of the great minds of the twentieth century. ... Read more

Reviews (1)

Excellent material that fits lots of class uses
A summary of his statement on p. 125 on "Russell's Mathematical Logic" describes the "vicious circle principle: forbids a certain kind of circularity which is made responsible for the paradoxes. The fallacy in these, so it is contended, consists in the circumstance that one defines (or tacitly assumes) totalities, whose existence would entail the existence of certain new elements of the same totality, namely elements definable only in terms of the whole totality." This led to the formulation of a principle which says that "no totality can contain members definable only in terms of this totality, or members involving or presupposing this totality." (The vicious circle principle). (Also a "not applying to itself principle to keep the vicious circle principle from applying to itself p. 126

In describing Russell's theory of types he says, "The paradoxes are avoided by the theory of simple types which is combined with the theory of simple orders - a "ramified hierarchy""

Godel argues that the vicious circle principle is false rather than that classical mathematics is false.

p. 202 "A remark about the relationship between relativity theory and idealistic philosophy (1949a) (Note that this view supports my usual presentations in class on this!)

"The argument runs as follows: Change becomes possible only through the lapse of time. The existence of an objective lapse of time 4, however, means (or, at least, is equivalent to the fact) that reality consists of an infinity of layers of "now"

p. 203 which come into existence successively. But, if simultaneity is something relative in the sense just explained, reality cannot be split up into such layers in an objectively determined way. Each observer has his own set of "nows", and none of these various systems of layers can claim the prerogative of representing the objective lapse of time. 5" ... Read more

Book Description

A long-awaited revision of one of the most respected primers on applied mathematics

Principles of Applied Mathematics provides a comprehensive look at how classical methods are used in many fields and contexts. Updated to reflect developments of the last twenty years, it shows how two areas of classical applied mathematics-spectral theory of operators and asymptotic analysis-are useful for solving a wide range of applied science problems. Topics such as asymptotic expansions, inverse scattering theory, and perturbation methods are combined in a unified way with classical theory of linear operators. Several new topics, including wavelet analysis, multigrid methods, and homogenization theory, are blended into this mix to amplify this theme.

This book is ideal as a survey course for graduate students in applied mathematics and theoretically oriented engineering and science students. ... Read more

Reviews (2)

Excellently organized book.
This book presents various mathematical principles in an organization I have not seen before. It starts with the idea of a transformation, then goes on to relate eigenvalues and eigenvectors to general spectral theory, explain how the need for closed function spaces naturally leads to Lebesgue integration (I know about Lebesgue integration before but I didn't know why it was needed), and show how the definition of certain inverse operators leads to distribution theory. This is a very natural way of organizing these principles. While other books, such as Strang's Intro to Applied Mathematics and Rudin's Real & Complex Analysis, provide you with one mathematical "toy" after another (Fourier series, Lebesgue integration, etc.), Keener's book tells you why you need the toy before giving it to you.

Am I stupid or this book is too advance???
This book doesn't clearify anything for you. No examples, No further explanation. It only keeps introduce various theories to you. It can compact theories that other books take 2-3 pages to explain it into 5 lines! I think you can imagine. Obviously, this book is not suitable to be your first book (not the second also). Buy it if you are sure that you are smart enough to understand it! ... Read more

Book Description

Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientsts, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continum physics.This compendium contains material that was previously available only in specialist journals.This is likely to become the standard reference work for all those interested in the subject. ... Read more

Reviews (1)

An easy call
Very simply: if you want to know a very great deal about topos theory, buy this book. I mean, seriously, if you plan to make real work on topos theory a part of your life, then grit your teeth and come up with the money. If you do not want to know a very great deal about it, do not buy this book. You can use it at the library as a reference.

If you merely want a professional understanding of what topos theory is, then read Johnstone's earlier TOPOS THEORY. That far shorter book gives a better overview. My Amazon review of it discusses others on the subject. Most are more accessible than Johnstone's books and go more into particular aspects of the theory.

This book is a reference on all the methods, and the latest results, in topos theory. If you want the definition of "split opfibration", it is here, along with some 80 pages of background, examples, and motivation. Johnstone does an heroic job of unifying the terminology and organizing the theorems.

More than that, Johnstone has written down an expert, encyclopedic view of the subject today. It is rare for a top mathematical researcher to give so deep an account of their field. It is rare for anyone to even work out such an explicit, coherent, extensive account of the whole. Not everyone will agree with his view. Some would like to see much less of such logical topics as "allegories", others would like to see the logic more formalized from the start. But Johnstone builds a case for his choices: partly implicit in his success at explaining things this way, and partly by explicit reasons.

If you want to know that much about the subject then you want to immerse yourself in this book. ... Read more

Book Description

The focus in this Second Edition is again on logistic regression models for individual level data, but aggregate or grouped data are also considered. The book includes detailed discussions of goodness of fit, indices of predictive efficiency, and standardized logistic regression coefficients, and examples using SAS and SPSS are included.

• More detailed consideration of grouped as opposed to case-wise data throughout the book
• Updated discussion of the properties and appropriate use of goodness of fit measures, R-square analogues, and indices of predictive efficiency
• Discussion of the misuse of odds ratios to represent risk ratios, and of over-dispersion and under-dispersion for grouped data

Updated coverage of unordered and ordered polytomous logistic regression models. 

Reviews (3)

Excellent Over view
Prof Scott Menard must be commended for writing an excellent book on Logistic Regression. Explaining it in the context of commercially available software packages is a very good idea. I was able replicate some his analysis using SAS on the data set used in this book (available on line from ICPSR, Univ of Michigan).

I eagerly await the next edition of this monograph. Thank you!

A Nice Overview
A good, cheap overview of logistic regression analysis.

I bought and I'm glad I did, but I don't refer to it like I do Hosmer and Lemeshow's text.

Excellent Guide to Logistic Regression
As its title suggests, this book is an excellent guide to using logistic regression in data analysis. I purchased this book because I needed to do several logistic regression runs for my dissertation. It turned out to be an extremely useful book for two reasons. First, it presents logistic regression alongside more traditional ordinary least squares (OLS) models. Therefore, if you already have a good understanding of OLS models, this book is very easy to follow. Second, its discussion of logistic regression issues in the context of SPSS or SAS makes it very easy to follow along with your own data analysis as you move through the book. Since statistical packages are always improving, this does date the book a little. However, this is a very minor concern. I believe Dr. Menard is to be commended for including issues regarding popular software packages in this work.

When compared to SAS's documentation, this book's greatest advantage is explaining in english (rather than mathematical notation) the assumptions and limitations of SAS's (and SPSS'S) algorithms. Its chapter on logistic regression diagnostics is alone worth the price of the book. In short, if you need to use logistic regression analysis and you already understand OLS, you cannot go wrong with this book. ... Read more

Book Description

The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as:* Is the Mandelbrot set decidable?* For simple quadratic maps, is the Julia set a halting set?* What is the real complexity of Newton's method?* Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps?* Is the Hilbert Nullstellensatz intractable?* Is the problem of locating a real zero of a degree four polynomial intractable?* Is linear programming tractable over the reals?

The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. ... Read more

Reviews (1)

blew me away
A warning about the title. Some confusion may arise over whether the book is about "computational complexity theory" or the field of "complexity" being pioneered by places like the Santa Fe institute. Without neccessarily pidgeonholing the book into one of these fields, I will warn "complexity" types that it dives heavily into the rigorous field of computational complexity theory (i.e. P/NP, theoretical upper bounds on running times of algorithms ,etc), and re-assure readers from the computational complexity theory camp that the book is more rigorous then the cover, or the title might lead you to believe.

My first introduction to this book/subject area was when Lenore Blum (one of the authors) gave a talk at Carnegie Mellon University, mostly following the outlines of the book. I found the talk to be so interesting that I went out and bought the book. While I am not a professional CS theorist, I did attend many of the theory seminars at CMU while I was an undergrad there (you may call me a "hobby theorist"). The talk on this book was one of the few that seemed as novel and mind-blowing to me as my first introduction to theory had been (just in terms of "Wow this is cool!" "Ooh, I never thought of those things in that way", etc).

The book is about a novel approach to applying discoveries from complexity theory to the analysis of numerical algorithms. Pure complexity theory quickly becomes unwieldy, as input/output sizes for real-numbers approximated on a turing tape depends on many factors, including the precision of the representation, and the representation method itself. Techniques from applied algorithms (most notably, the "RAM machine" model of the 1970s) have the unfortunate side-effect of being able to solve problems in NP in polynomial time. Blum (and the other authors) take the novel approach of just allowing this side effect, while getting meaningful complexity bounds on real-valued computation, by creating a real-valued analog to the discrete turing machine used in classical complexity theory. Along the way, the authors show that, while this model does allow problems in NP to be solved in polynomial time, it introduces a class, which is not NP, but analogous to it, in the sense that theorems on real-valued algorithms have similar proofs to their discrete counterparts in classical complexity theory.

While this is not neccessarily useful to most practical programmers, it is, in addition to being a fascinating and novel way to look at numerical algorithms, also a fascinating subject to think about when looking at the physical world. Among the physical processes that can be looked at from the perspective of this book, are the much-hyped chaotic systems prevalant in the (unfortunately named -- and not very closely connected to "Computational Complexity Theory") field of "complexity" associated with the Santa Fe institute (hence the picture of the Mandelbrot set on the cover, which the authors study as a decidability problem within the framework of the new real-valued Turing machines introduced in this book)

My one complaint about the book was that, while the talk Lenore gave at CMU was aimed at an audience more familiar with computational complexity theory then with continuous mathematics, the book goes the other way around, making painstaking explanations of elementary computability / complexity theory, but assuming a strong knowledge of continuous mathematics. ... Read more

Book Description

The second edition of the popular A First Course in Fuzzy Logic will continue to provide the ideal introduction to the theory and applications of fuzzy logic. The authors provide a firm mathematical basis for the calculus of fuzzy concepts-necessary to design intelligent systems-and give the student a solid background for further studies and real-world applications.This new edition provides many new exercises designed to enhance the reader's understanding of the concepts. The authors have expanded on the algebra background needed for the more advanced topics, and include significant new material on basic connectives and the algebraic properties of fuzzy logic, rough sets, conditional events, distributions of random sets, and derivatives of fuzzy measures.With its comprehensive updates, A First Course in Fuzzy Logic, Second Edition presents all the background necessary for students to begin using fuzzy logic in its many-and rapidly growing-applications. ... Read more

Reviews (2)

A First Course in Fuzzy Logic
If you are looking for a book on Fuzzy Logic Theory, this is a good book. It explains the mathematical basis of fuzzy logic, the mathmatical symbols used, provides proofs for theorems, etc. The chapters and sections are clearly labeled in the table of contents, making it easy to zero in on a topic. It is a College Text Book, with problems to solve at the end of each chapter. Selected problems have answers in the back of the book. Unfortunately, that wasn't the type of book that I expected or needed. I was looking more for the hands on, how to, type of book. That is the only reason it did not get 5 stars.

Outstanding, for people who are interested in this area
This is the clearest explanation and application of fuzzy logic that has been published. It is thorough, without being arcane or pedantic. ... Read more

Reviews (1)

Important Contribution to Modern Epistemology
The 20th century saw more advances in knowledge than could filter down to general society. Relativity and Quantum Theory are part of the vernacular, even if the popular conceptions are not necessarily good generalizations of their counterparts in science. The corresponding advances in philosophy, however, have stayed more in the province of academia, largely because philosophy itself has become highly technical; but the physics of beyond-everyday-experience have demanded these advances, primarily in epistemology, because the fundamental questions of science today are of meaning and understanding.

Understanding the Infinite is a work of epistemology. Its contribution to the foundations of general knowledge demand that it disseminate beyond academia, although the ground Lavine breaks requires the extensive citations and technical style he employs. The author poses and addresses the following question. If set theory is so intuitively self-evident and seemingly such a fundamental underpinning of all mathematics, why is it so hard to express technically and why has the axiomatization of set theory been so controversial? Set theory was the big idea which the mid-20th century educational establishment thought important enough to indoctrinate schoolchildren with in the guise of new math. Yet set theory never took root in popular consciousness, certainly not the notion of transfiniteness.

Lavine starts out by dispelling the anecdotal account of the development of set theory, which has misled even professional mathematicians and philosophers to conclude "The fundamental axioms of mathematics...are to a large extent arbitrary and historically determined." He constructs what he claims is the correct historical development of set theory (I'll let historians of mathematics decide this) including sidetracks into Russell's failed program to equate mathematics and logic (and in the process dispels the significance of Russell's paradox), and von Neumann's axiomatization of set theory emphasizing functions. The outcome of his exposition is the Zermelo-Fraenkel axiomatization with the Axiom of Choice (ZFC), today's common form of set theory. These chapters by themselves could serve as an introduction to set theory, except that the Continuum Hypothesis is barely mentioned, since it plays no role in Lavine's program. Admittedly, he has nothing new to add.

The main event is Lavine's epistemological tour-de-force. Building upon work of Jan Mycielski he introduces the reader to the concept of finitary mathematics and constructs a finitary ZFC, showing that this theory justifies the adoption of what he calls the "Axiom of Zillions" (indefinitely large sets) in which we have access to very large sets' ordinal, but not necessarily its predecessors. The final step is to show this all "intuitively" extrapolates to ZFC.

QEF, QED.

I introduced physics in the opening paragraph of this review because I see Lavine's rigorous treatise in the epistemology of mathematics as a contribution to the grand unification of physics, mathematics, and epistemology. Lavine treads lightly in the physical realm. He writes "...modern physics makes it seem likely that the physical universe is of finite extent..." All of the dominant cosmologies put forth in the 20th century incorporated this misdirection set off by general relativity. On a large scale the universe must be curved. Ironically Lavine published in 1994, just as new astronomical observations began whispering "in three dimensions the universe is Euclidean". If that whisper becomes a shout in the 21st century, as appears likely from the mounting evidence, physics will have to address the transfinite.

The Calculus had to be put on a firm theoretical foundation so that it could be used as a tool to advance knowledge without justifying its use. We may see that Lavine's epistemology will do the same for set theory and transfinite numbers. ... Read more

Reviews (12)

An excellent inroduction to computational complexity
Papadimitriou is one of the great minds in computer science, which is reflected in this gem of a book. His prose is very engaging and he covers just about every topic (although be it lightly) relevant in modern complexity theory without overly diluting the proofs and results (for example, he gives a nice concise proof of Razborov's theorem on monotone circuits).

Good overall.
A well-written book that teaches you how to think about complexity theory instead of just a flat summary of results. Something like Lewis and Papadimitriou's _Elements of the Theory of Computation_ would be more than enough preparation for this (note that the style of these books is quite different- this one is more informal and descriptive). Covers all the material you need in a first text. Has a good little introduction to mathematical logic in it, including a nice succinct version of Godels Incompleteness Theorem. Lots of interesting exercises.

Upon Leaving The Realm Of Sense, For Points Distant
*Computational Complexity* is a great introduction to contemporary formal logic, which is more heavily mathematical than even the great mathematical logicians of the past were. It's a great introduction, even though they put a good-lookin' chick on the *back* cover (what's that about?), because Papadimitriou does not suffer from what Quine called *mathematosis* -- he is not assimilating the material of logic (anything and everything symbolic) to the rigorized abstractions of mathematics. And as such, this book defines what I think *computer science* should be today by choosing the "greater of two evils". If you think that computer science began with Turing like most people, CS is about building machines that do what thinking beings do (problems of detail); but if you think that computer science began with Church's undecidability theorem, CS would be about figuring out why thinking beings fall down on the reasoning job (troubles with principle).

I prefer the second definition; and although I'm a little old-fashioned in my tastes (prove it by me), this book demonstrates such an attitude can be forward-looking. Although Church is not venerated throughout the book, a task handled by Papadimitriou in his earlier CS introduction with Lewis, unlike Hopcroft and Ullman the spirit of Church is very much present in Papadimitriou's teasing-apart of complexity problems from applied CS. Yes, it's never about the physical machine, and Ryle can go away instead of work like this -- which in my opinion could form the basis of a "computational psychology" concerned with the will to truth rather than the will to power.

All in one roof, but presentation very poor
I agree with the review by Arthur Fischer. Papadimitriou might
be an excellent researcher, but his communication skills are
hopeless and horrible. The typos make learning even harder.

Perhaps someone like Michael Sipser should take up the task of
rewriting this book.

Be careful, but it is worth reading
Yes,it is generally "hard" for undergradute students even grad. students. If you are taking course "Theory of computing", I would like to recommend Sipser's or Cohen's books for introduction's supplement. However, this book covers too many topics, therefore becomes too dense, so that you should read it carefully. I admit that I could not go through more than 20 pages of this book in the beginning. But I was keeping on reading and surveying some "easy books". Finally, I understand most parts of this book. However, if some readers want some more advanced and recent topics. They should refer to some conference's papers hosted by ACM or IEEE. ... Read more

Book Description

This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with the fundamentals of mathematical language and proof techniques (such as induction); then applies them to easily-understood questions in elementary number theory and counting; then develops additional techniques of proofs via fundamental topics in discrete and continuous mathematics. Topics are addressed in the context of familiar objects; easily-understood, engaging examples; and over 700 stimulating exercises and problems, ranging from simple applications to subtle problems requiring ingenuity.ELEMENTARY CONCEPTS. Numbers, Sets and Functions. Language and Proofs. Properties of Functions. Induction. PROPERTIES OF NUMBERS. Counting and Cardinality. Divisibility. Modular Arithmetic. The Rational Numbers. DISCRETE MATHEMATICS. Combinatorial Reasoning. Two Principles of Counting. Graph Theory. Recurrence Relations. CONTINUOUS MATHEMATICS. The Real Numbers. Sequences and Series. Continuity. Differentiation. Integration. The Complex Numbers.For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics. ... Read more

Reviews (2)

The best reference for Proofs
This is an advanced book, with a lot of information on every page. I use it as a reference book, since it has hundreds of wonderful proofs and problems, along with thorough and concise definitions for just about every major branch of mathematics.

It's highly recommended for anyone who is *serious* about mathematical proofs. Although the book is packed with material, it's a small book, so it's one of the first I choose to take with me when I travel.

Very good introductory text
With a disposition towards problem solving, the book was very helpful for someone like myself who needs the mathematical tools but lacks the background. Occasionally the book lapses into the silly notion that you've seen some of the material and concepts before. ... Read more

Book Description

Janos Bolyai (1802-1860) was a mathematician who changed our fundamental ideas about space. As a teenager he started to explore a set of nettlesome geometrical problems, including Euclid's parallel postulate, and in 1832 he published a brilliant twenty-four-page paper that eventually shook the foundations of the 2000-year-old tradition of Euclidean geometry. Bolyai's "Appendix" (published as just that--an appendix to a much longer mathematical work by his father) set up a series of mathematical proposals whose implications would blossom into the new field of non-Euclidean geometry, providing essential intellectual background for ideas as varied as the theory of relativity and the work of Marcel Duchamp. In this short book, Jeremy Gray explains Bolyai's ideas and the historical context in which they emerged, were debated, and were eventually recognized as a central achievement in the Western intellectual tradition. Intended for nonspecialists, the book includes facsimiles of Bolyai's original paper and the 1898 English translation by G. B. Halstead, both reproduced from copies in the Burndy Library at MIT. ... Read more

Book Description

Begun in Hungary in the nineteenth century, Mathematical Olympiads are now held for high school students throughout the world. They feature problems which, though they require only high school mathematics, seem very difficult because they are unpredictable and have no obvious starting point. This book introduces readers to these delightful and challenging problems and aims to convince them that Olympiads are not just for a select minority. The book contains problems from the British Mathematical Olympiad (BMO) competitions between 1965 and 1996. It includes hints and solutions for each problem from 1975 on, a review of the basic mathematical skills needed, and a list of recommended reading, making it an ideal source for enriching one's experience in mathematics. ... Read more

Reviews (3)

Very usefull book.
I teach students, a mathematical olympic team, and this book will help them to get the training they need in order to have more medalls and achievements in their carreers. It has everything they need to do their exams and practice.

Excellent introduction to problem solving
The last time I competed in a mathematical competition was in elementary school. So when I recently picked up and tried to solve some old Olympiad and Putnam questions I was dumbfounded, to say the least. Then I discovered "The Mathematical Olympiad Handbook." This book is an excellent intro to problem solving. It is an immense help in dealing with the frustration of problems that will not yield. No solutions are given-- only outlines to the solutions(which you should only read part of if you get stuck, as the outlines will easily lead to a solution). This structure was an excellent choice as it gives novice problem solvers a chance to solve problems on their own with a little help if they need it. Highly recommend!

[people new to problem solving should also check out "The Art and Craft of Problem Solving" by Paul Zeitz]

A must-buy for problem-solvers.
This book has proven: The olympiad problems are not so hard as they appears. For every problem, there is a valuable solution outline ,teaching us how to attack it in a natural way. Furthermore, here is a good list of reference books. ... Read more

Book Description

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists. ... Read more

Reviews (1)

Excellent book.
Excellent book. The best in its field. I would recommend it, particularly for students. ... Read more

Reviews (1)

Excellent introduction and something more
This book presents an excellent introduction to the subject, but also goes beyond that, presenting with a fair amount of the detail the theory of Priestley representation. The excercises start at the basic level of checking the understanding of definitions, allowing the reader to build confidence out of the practice. The fact that Priestley herself co-authored it is definitely a plus. ... Read more

Book Description

This long-awaited book replaces not one but both of Hughes and Cresswell's two previous classic studies of modal logic: An Introduction to Modal Logic and A Companion to Modal Logic.A New Introduction to Modal Logic is an entirely new work, completely rewritten by the authors to incorporate all the new developments that have taken places since 1968 both in modal propositional logical and modal predicate logic, but without sacrificing the clarity of exposition and approachability that were essential features of the earlier works. The book takes readers through the most basic systems of modal prepositional logic right up to systems of modal predicate with identity.It deals with both technical developments such as completeness and incompleteness, and finite and infinite models, and discusses philosophical applications, especially, in the area of modal predicate logic. ... Read more

Reviews (3)

Pleasant and Thorough
This book is the best introduction to Modal Logic I know, and it is great for reference too. It gives a general survey of modal propositional and even modal predicate logic (this latter is pretty difficult and rarely presented in introductory books). There is one feature of this book that I really love: the authors try to present everything without supposing anything on the part of the readers. I really suffer when I get a book that supposes that strange type of pre-requisite that appears to be everything minus the subject that is being presented - and sometimes the author even supposes that you have already met his subject before! Hughes and Cresswell write in a pleasant pace, sometimes repeating themselves, but this is good for educational purposes. I like their analogies: the presentation of K-validity as a modal game is really superb. They do not exagerate with mathematical formalisms, using only when indispensable, and this is good for the non-mathematician. I recommend it for every reader interested in logic in general, mathematicians or philosophers.

slow and painful
If you have never studied logic, this is the book for you. But if you have, find something else. This book is slow and it spends too much time on the deductive system in propositional logic. Tedious.

OK, but Hard to Wade Through
Hughes and Cresswell (sounds like a deli sandwich, unfortunately) have written what is probably the best introduction to modal logic on the market. They are clear, orderly, organized--but they don't provide the student with anything to make his or her foray into the subject any more pleasant. It's a text for logicians-to-be with excessive-compulsive disorders; that would be most, but not everyone. The authors would do well to reformat their text, but the content is superb. ... Read more

Reviews (1)

Best intro to fuzzy logic
One thing frustrating with fuzzy logic books is that they are full of weird notation and have little or no adequate examples to illustrate the concepts. You won't find that problem with Tanaka. He takes you by the hand and shows you all the basics using easy to understand language. He provides enough examples that illustrate every concept. This book is very readable and I am willing to bet that an average high school student can find it accessible. It's that good. ... Read more

Book Description

(Reprint of the 1967 edition) ... Read more

Reviews (3)

great book
There are a lot of good introductory books on computation theory,
but after reading them you may be left asking "okay, what do
I read next?" Well _this_ is the book. If you're looking for something in between the undergraduate intro books and
the research-level articles then this is for you. It develops recursive function theory in a succinct, mathematically mature manner that is freed from the details of any particular formalism. You should have previous exposure to turing machines and undecidable problems, an appreciation of the defense and use of Church's thesis, and familiarity with basic mathematical logic. Just to be clear, this book is NOT:
-a computer science, programming, or algorithms book
-an introductory book
-a book about automata or weak models of computation (such as regular languages or context-free grammars)
-a complexity theory book (no time bounds or np-completeness etc)

A classic!
The definitive book on computabilty and recursive function theory. I remember reading this book in preparation for research in complexity theory. I found it very stressful reading the book, but it was a good kind of stress. The kind that forces you to think deeply about what the author is writing about. In addition to the main text, the author provides numerous thought-provoking problems whose study could make a coure unto themselves. I read this book as a 3rd-year graduate student in

math. If you are an undergraduate and are interested in computability theory, I recommend Nigel's Cutland's book on the subject.

Good reference
I looked long and hard for a reference in recursion theory and this was the only one which was acceptable. Luckily it is also quite good.

Most books in the subject either introduce the material in their own non-standard notation which, while suitable for a survey course in the material is of little help when attempting to actually read papers in the field. These books are also usually very basic ignoring things like the arithmetical hierarchy. Other books in this subject seem to mostly be advanced texts and don't cover, or cover very briefly, the important theorems.

This book starts at turing machines and recursive functions. Going through the basic results like the halting problem and rapidly moving on to more advanced topics like creative sets, cylinders and hypersimple sets. Posts problem(with Friedberg's solution) and the fixed point theorem are covered as well. The final part of the book covers degrees of unsolvability arithmetical hierarchy and the analytic hierarchy.

While the book does cover recursive fucntions and turing machines I would suggest previous experience with them before reading as the coverage is brief and doesn't give the reader a feeling of how these systems work.

If you are taking a class in the subject or want to understand modern recursion theory this is a wonderful place to start. ... Read more

Book Description

Set Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as a desktop reference. ... Read more

Reviews (4)

A good but unreadable book
It's really a good book for researchers in set theory. But it is NOT an introduction for students who want to know what is set theory. You will feel you are so stupid if you read this book without any set theoretical background. I recommand Kunnen's book for those people who are interested in set theory but have no any (or only a little) set theory knowledge.

This edition has been completely revised!
Just wanted to point out that all the reviews here dated before Feb 2003 are referring to older editions. The new one has been totally revised (no laundry list of corrections at the end) and also expanded -- lots of material from the last 25 years of set theory research is now included. Most notable among these is material on proper forcing and pcf theory. (There is even a section on my research interest, mutually stationary sets, and this is a notion which was just published for the first time 2 years ago!) The book is still just as informative and readable as the previous editions.

A classic
This book is a wonderful reference volume for set theory. It contains a clear and readable explaination of all the things a set theorist needs to know. I have only one complaint: "revised edition" simply means that a 20-odd page errata has been appended....

My favorite book on set theory.
In 1979, I was a first-year graduate student in mathematics. One summer day, I was looking in the math section of Stanford bookstore and saw this thick green volume with the simple title Set Theory (by Thomas Jech). I couldn't help pulling the tome off the shelf. I flipped through the pages in awe. This book had everything about mathematics that I had always wanted to know.

After about an hour, I reluctantly looked at the price and it was just too much; I had to put it back on the shelf. But for the next month, that book was all I could think about. I finally went back and bought it.

Two years later after hooking up with my adviser and embarking on research in set theory, I started working through Jech's book starting on page 1. It took me 2 years to work through the entire book, and for much of that time I had the opportunity to present what I was learning in seminars.

That book is a real treasure. I don't think I've spent as much time poring over any other book. I think the presentation of material is fantastic and the coverage is thorough (or it was at the time I studied it--probably his recently updated work also has this attribute).

I would recommend this book (or rather the most recent edition of it) to any serious graduate student specializing in set theory.

Two areas where I needed supplementary study were in his approaches to the constructible universe and to forcing. These are important areas, and Jech does a fine job in his approach, but certain approaches other than his have become more of a standard, and any serious researcher will have to become familiar with these standards. Jech uses Boolean algebras (primarily) in his development of forcing (and his development is excellent) whereas by now, the usual approach is with partial orders. Also, Jech develops L as a transitive model that is closed under "Godel operations"--a perfectly valid approach. These days, though, the formula-based approach is more common in the literature.

Nonetheless, Jech's wide variety of forcing applications, his in-depth treatment of large cardinals, and his compact surveys of saturated ideals and descriptive set theory make his work really an outstanding contribution. ... Read more

Book Description

An essential reference for anyone grappling with advanced mathematics, this Fourth Edition help readers master the basic techniques that are used in all proofs, regardless of the mathematical subject matter in which the proof arises. Once the reader has a firm grasp of the techniques, they’ll be better equipped to read, understand and actually do proofs. They’ll also learn when each techniques is likely to be successful, based on the form of the theorem.

"The book covers all the basic proof techniques in a very readable, concise way without overwhelming the student. The organization is great. I like the short chapters highlighting only one concept at a time." (Josephine Hamer, Western Connecticut State University)

"Very clear, rigorous, extremely thorough, almost unique in what it tries to do, reaches out to weaker students." (Michael Thaddeus, Columbia University) ... Read more