Book Description

Reasoning under uncertainty, that is, making judgments with only partial knowledge, is a major theme in artificial intelligence.Professor Paris provides here an introduction to the mathematical foundations of the subject.The author presents the key results on the subject, and formalizes within a unified framework the main contemporary approaches and assumptions. He concentrates on giving clear mathematical formulations, analyses, justifications, and consequences of the main theories about uncertain reasoning. ... Read more

Book Description

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements. Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)? ... Read more

Book Description

Reviews (5)

Knuth is Analysis of Algorithms
Befor Donald Ervin Knuth, there was no such thing as the Analysis of Algorithms. He is a visionary in this field, and these selected papers are a testomony to his greatness in this field.

Not as Approachable as Knuth's Other Writings
To a reader interested in the analysis of algorithms, this book doubtlessly contains a number of very thorough analyses, with some interesting correspondence about the origin of complexity O()-notation and the term "NP-hard" thrown in.

As a reader more interested in Knuth's work in general, I must admit that, despite having advanced degrees in CS and a quite decent math education, I found myself unable to follow the heavy mathematics employed, and ultimately, the problems analyzed were not interesting enough to me to make me spend the effort to follow it. One of the chapters was titled "A Trivial Algorithm Whose Analysis Isn't", and this probably sums up why I'm not entirely enthusiastic about this book and about the field of algorithm analysis in general.

Knuth wouldn't be Knuth if he didn't throw some lighter material in as well, and chapter 2, "The Dangers of Computer Science Theory", was quite amusing.

Knuth++
Professor Knuth collects a charming and valuable set of papers into one terrific place. Don't miss his satirical "anti- Science" article "The Dangers of Computer Science", Chapter two, first presented in 1971.

Reference manual of it Analyzes of Algorithms
Donald Knuth is recognized as guru of the science of the computation. This publication comes to already complement its collection consecrated. The Analysis of Algorithms is one of the most important areas of the science of the computation. Excellent release.

Select Papers on the Analysis of Algorithms
Donald Knuth is recognized as guru of the science of the computation. This publication comes to already complement its collection consecrated. The Analysis of Algorithms is one of the most important areas of the science of the computation. Excellent release. ... Read more

Book Description

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. ... Read more

Reviews (1)

Platonism=Fictionalism=Realism?
This is a remarkably forceful and ambitious book but a very
worthy read nonetheless. Balaguer is clearer in his arguments than about any other contemporary philosopher I have read on the subject! He does however make a few discernable mistakes and shows a surprising lack of depth is some of his tangential examinations as pointed by some of his reviewers (I am thinking of Colyvan and Zalta whose review of this book can be found on the web). Also, out of breath as I was, by the time I finished this book, I cannot say I feel persuaded by its thesis with respect to the indescernability between Fictionalism and Platonism. Still Balaguer's notion of Full-Bloodied Platonism, the peculiar point-of-view he develops and embraces in this work is extremely interesting and challenge: it comes down to the notion that all "broadly possible" mathematical structures exist. This happens to be, though Balaguer seems anaware of it, a thesis currently arrived at by physical cosmologists speculating about the "Multiverse" (see Mark Tegmark's recent Scientific American article on "Parallel Worlds")! When different lines of speculation arrive at the same concepts there is some hint of
historical consensus one tends to suspect a metaphysical corner
where we are all about to get stuck for a while! On the other hand I cannot help to remark how simplistic and misleading is
the language in which philosophers insist in carrying their arguments! An example from the beginning is the characterization
of an abstract object as one that exists "non-spaciotemporaly".
Though he ends up debating some of the obvious problems with this
distinction Balaguer never addresses today's scientific consensus that space-time itself is an abstract object of some sort (except if you ask Julian Barbour and his Leibnitzian crowd), either Riemann space or Multiverse, so one may naturally ask why should it be a previledged reference for existence? On this matter I take a a more radical view than Balaguer, which I would call Full-Bodied Platonism, by arguing that all that exists are abstract mathematical objects. What he
calls spaciotemporal existents (that includes us, at least the ones among us who cartesianly think they exist) whose existence is merely contingent on our participation in the true (eternal and necessary) existence of such abstractions. (But wait! Isn't that what Plato thought?) ... Read more

Book Description

This is the first of a three-volume collection of David Lewis' most recent papers in all the areas to which he has made significant contributions.This first volume is devoted to Lewis' work on philosophical logic from the past twenty-five years. The topics covered include: deploying the methods of formal semantics from artificial formalized languages to natural languages, model-theoretic investigations of intensional logic, contradiction, relevance, the differences between analog and digital representation, and questions arising from the construction of ambitious formalized philosophical systems. ... Read more

Book Description

Nonmonotonic reasoning in its broadest sense is reasoning to conclusions on the basis of incomplete information. Given more information, previously drawn inferences may be retracted.Commonsense reasoning has a nonmonotonic component; it has been argued that almost all commonsense inferences are of this sort. From the end of the 1980s to the present there has been an explosion in research in nonmonotonic reasoning.It is now possible to understand more clearly the properties of the major formalisms from a metatheoretical point of view, the relationships among the formalisms and their connection to independently developed proof methods.The goal of this monograph is to make this understanding more accessible. ... Read more

Reviews (1)

Fuzzy Translation for Logic Beginners
I found this text to be a satisfactory introduction to the principles of Fuzzy Logic. I don't think that the book presents the relevant information to developing a fuzzy system, but rather tries to excite the viewer with useless examples of fuzzy systems.

It did answer and explain in detail fuzzy logic and the history behind the controversial mathematical concept. To predict the characteristic of something based on a table of measured values is something whose problem lies in the implementation of such a process. It would take a good amount of testing to find how much torque is needed to counteract the steering in commuter trains. The previous sentence was one of the examples used in the text. ... Read more

Book Description

Domain theory is the mathematical framework that is used to model the semantics of computer programs and the theory of computation.This is the first book on the subject that attempts to provide a rigorous introduction to the topic in a manner accessible to computer scientists by motivating the mathematics with computer science examples. ... Read more

Book Description

The reader of this book is probably about to teach or take a "first course in proof techniques." Students are taking this course because they like mathematics, and the authors hope to keep it that way. At this point, they have an intuitive sense of why things are true, but not the exposure to detailed and critical thinking necessary to survive in the mathematical world. The authors have written this book to bridge the gap. Often, students beginning this course have little training in rigorous mathematical reasoning; they need guidance. At the end, they are where they should be; on their own. The authors aim is to teach the students to read, write and do mathematics independently, and to do it with clarity, precision, and care. ... Read more

Book Description

This book is a modern introduction to model theory which stresses applications to algebra throughout the text.The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications.The author also includes an introduction to stabilitytheory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories.One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry.David Marker is Professor of Mathematics at the University of Illinois at Chicago.His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry.This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998. ... Read more

Reviews (3)

A very good book
There are several good introductory model theory texts out there today, and it seems that each one is strong where another is weak. To get a gap-free introduction to the subject, you best get them all--sadly, for many of us this is not possible.
Anyway, you want to know where is Marker is strong. Most notably, his book has plenty of exercises (ranging as they should from trivial to challenging) and many good examples. You get good at (anything really) model theory by doing it, not just reading about it. You can't get good at mathematics by working in a vacuum, so you need to see examples. Marker's book succeeds by providing both of these things, and any book that doesn't you should buy last (sadly, Poizat's book, though good for what it is, has no exercises). Marker's pace is quick: the first two chapters give you the logical toolbox, chapter three eliminates quantifiers, in chapter four you construct skinny models (omit types) and fat ones (saturation/realizing types), in chapter five you do some combinatorics, six wraps up classical model theory (Morley's categoricity theorem) and gives the groundwork for stability and modern techniques (pregeometries, forking etc.). The rest of the book gives good coverage (for the space allotted) of algebro-geometric ideas that are pretty current. Marker's chapters seem a bit more application oriented than Hodges' (this is a hard call), and his book does give more information about current topics (stability, algebra, geometry etc.). Contrary to the opinion of another reviewer, Marker's book is not obscure. The subject often exhibits a high degree of abstraction, but through his use of examples, Marker is good at guiding the reader's intuition and giving a sense of "mathematical relevance" to the subject (but see below).
You also want to know what the (possible) weaknesses are. There are some typos-this is actually not so trivial a weakness if you think about it. Many interesting older ideas that you can find in Chang and Keisler or Hodges are absent (maybe because you can find them in those books), e.g. Craig interpolation, formula preservation theorems (Marker leaves a couple as exercises), ultraproducts. One response may be that the first omission is better called "logic", the second "somewhat pointless", and the last "set theory". Finally, overemphasizing examples can misdirect you-the way a theorem applies to particular cases often masks the "real" sense of the theorem. I'm not saying Marker overemphasizes examples, but some of you love pure generalities, others love concrete matters...all this may be a matter of taste.
How does Marker's book stack up against other's? Similar scope as Poizat's, but has exercises and is less chatty. Marker's is better (for you). More up to date than Chang and Keisler's, but weaker in explicating the logic (*in the chapters* that is, there are nice logic problems in the exercises, but you are pretty much *on your own* when you do them), and Chang and Keisler are easier to read. They are more fun to read than almost anyone else though. Marker's book is still better for the modern student. If you're serious, you'll have to get C&K eventually anyway if only because of what their book is to the subject. Hodges is friendlier, stronger in the logical topics, "cooler" in some way (hard to describe...he's witty and works slickly), but Marker delves into some more contemporary ideas. It's a toss up. If you don't know anything about model theory, get Hodges' (also one of the *dinky-though-probably-handy* books that you can find on this site-Doets' book?). If you know some stuff and want to see modern applications explained well, get Marker...Actually, they're both affordable and very good. Save up and get both.
Does the book count as an introduction? Unhappily for the beginner, yes. Time was you could talk forever about Skolem's theorems, do simple diagram-chasing proofs of things that proof theorists knew in the 1930's, prove things with monstrous ultraproducts instead of quoting the compactness theorem and still say you were doing model theory. But times have changed and so have interests. Marker's book reflects this change well.

Great book
While the previous generation of model-theoreticians may have learned their craft from Chang-Keisler, the graduate students of the future will undoubtedly gravitate towards Marker's introduction. It may not be as fun to read as Poizat's, but it is certainly an extraordinary reference for the serious beginner. If you are studying for comprehensive exams on model theory--e.g., you are a graduate student at Wisconsin-Madison--this is the book you must read. Consider also checking out free class notes on the webpages of the experts. The logic group at the University of Illinois at Urbana-Champaign has some good resources.

deceptive title
Marker's book is not an introduction! It starts off obscure and stays that way throughout. No motivation is developed and only the highyl technical aspects of model theory are presented. He does not give in depth account to maximizing techniques and also glides over the lowenheim-skolem theorems, the proof of both totalling less than a page and a half! While this is not particularly important, no historical constext is given either. A positive aspect of this book is the relevance it has to modern algebra. Marker uses ample examples from algebra. In my opinion, this book is not an introduction, but a cross-disciplinary text for both model theory and modern algebra. Anyone seeking an actual introduction to model theory, or even an historical exposition of the subject need NOT purchase this book. ... Read more

Reviews (2)

Nope, didn't like it.
This book is an introduction to mathematical logic, covering the syntax and semantics of propositional and first-order logic, the Hilbert-style proof system and its completeness, some model-theoretic material, and Godel's (first) incompleteness theorem. Its more formal and rigorous than most introductory books, which is the style I prefer, but I was left feeling unsatisfied with the book. It was hard to nail down exactly what I didn't like about it, but what I came up with is this: although the theorems and proofs are ok, considered one at a time, the overall perspective of what's going on and how things relate to each other was left hazy. Perhaps better exposition and historical background would correct this, but I found the book unsuitable for self-study for a beginner. This was where I first learned Godel's incompleteness theorem, and even though the version presented is particularly weak (Peano arithmetic is incomplete), I was left confused about the significance of the theorem and exactly what assumptions were used in the course of the proof. I see now that their attempt at simplification is what led to my misunderstandings. If you're looking for a good general mathematical logic book, I seriously recommend you get Enderton instead (see my reviews). If you want a book focused on the incompleteness theorems get Smullyan's excellent GIT.

A good simplification for teaching
This is an interesting simplification of Goedel's first incompleteness theorem. The book assumes there is a standard model of the Peano axioms, so that in effect it assumes the axioms consistent and even true. Then it shows nonetheless the axioms cannot decide every sentence. This brings out the main point of incompleteness, I think. But it is far weaker than Goedel's proof in two ways: it uses stronger assumptions, and it proves incompleteness only for the standard Peano axioms. There is a brief discussion of how this kind of proof would work for any consistent extension of the Peano axioms but I did not find it very helpful to my class. Of course the assumptions pay off in a very much quicker proof and much less concern with syntax.

In fact, the main problem with the book is that the assumptions are never made quite clear. The authors say several times that their result is weaker than Goedel's, but never say why. A discussion of this somewhere in the book would be helpful--both to students and profs. Presumably they do not use the full strength of a standard model of the Peano axioms, but it would be a chore to go through and see just what they do use.

I taught this book as a one semester course for students who had previously seen predicate logic in an intro course. To fit it into a semester I skipped the chapter on model theory (not needed for the incompleteness theorem) and the material at the end on recursive functions. The book gives a very pretty account of induction, stressing from the start that the natural numbers are just one case of an inductive structure. This made later inductions on, say, well-formed formulas, very clear to the class. The students got the compactness theorem very easily, as they had not in other class I've taught from other books. The short account of non-standard models for arithmetic is helpful in showing that Goedel's theorem is *not* about whether the Peano axioms say all there is to know about arithmetic--the fairly simple compactness theorem already shows no first order theory can do that. I expect to use this book again the next time I teach the subject. ... Read more

Book Description

"Proofs and Fundamentals: A First Course in Abstract Mathematics" is designed as a 'transition' course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra, and real analysis.

This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material including sets, functions, and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics, and the Peano Postulates. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and writing are never compromised. The material is presented in the way that mathematicians actually use it; good mathematical taste is preferred to overly clever pedagogy. There is a key section devoted to the proper writing of proofs. The text has over 400 exercises, ranging from straightforward examples to very challenging proofs.

The excellent exposition, organization and choice of topics will make this text valuable for classroom use as well as for the general reader who wants to gain a deeper understanding of how modern mathematics is currently practiced by mathematicians.

Also by the author, "A First Course in Geometric Topology and Differential Geometry", ISBN 0-8176-3840-7. ... Read more

Reviews (1)

an excellent introduction to formal mathematics
This book is an excellent and thorough intorduction to the world of formal mathematics. Typically, a new mathematics student finds himself or herself picking up random mathematical concepts and techniques of proof along the way, without ever having the chance to sit down and go through the fundamentals of formal mathematics and proof. This book covers a little of everything, and has a thorough introduction to sets, functions, inverses, equivalence and order relations. There are also sections on introductory number theory, algebra, combinatorics, logic, and much more. It provides an excellent overview for the student who will be using these tools on a daily basis, for the layman who is interested in finding out what mathematics is really about, or for the seasoned mathematician who needs a good general reference book. There are also extensive and thorough sections on the construction and writing of mathematical proofs -- somthing on which many new and not-so-new mathematicians could use some improvement. ... Read more

Amazon.com

The search for infinity, that sublime and barely comprehensible mystery, has exercised both mathematicians and theologians over many generations. Jewish mystics, in particular, labored with elaborate numerological schema to imagine the pure nothingness of infinity, while scientists such as Galileo, the great astronomer, and Georg Cantor, the inventor of modern set theory (as well as a gifted Shakespearean scholar), brought their training to bear on the unimaginable infinitude of numbers and of space, seeking the key to the universe.

In this sometimes technical but always accessible narrative, Amir Aczel, author of the spirited study Fermat's Last Theorem, contemplates such matters as the Greek philosopher Zeno's several paradoxes; the curious careers of defrocked priests, (literal) mad scientists, and sober scholars whose work helped untangle some of those paradoxes; and the conundrums that modern mathematics has substituted for the puzzles of yore. To negotiate some of those enigmas requires a belief not unlike faith, Aczel hints, noting, "We may find it hard to believe that an elegant and seemingly very simple system of numbers and operations such as addition and multiplication--elements so intuitive that children learn them in school--should be fraught with holes and logical hurdles." Hard to believe, indeed. Aczel's book makes for a fine and fun exercise in brain-stretching, while providing a learned survey of the regions where science and religion meet. --Gregory McNamee ... Read more

Reviews (49)

An excellent combination of math, psychology, and Cabalah
This is easily the best book on mathematics this year. Amir Aczel has done it again, after Fermat's Last Theorem and God's Equation. Here he tackles one of the most difficult areas in mathematics--set theory--and weaves a very readable narrative including elements of Jewish mysticism and psychology. This book deals with the tormented life of Georg Cantor, the first person in history to understand the nature of infinity. Read it! I will say no more, so I don't spoil your enjoyment.

A delightful guide to the foothills of a huge subject
This delightful little book is a Cook's Tour of some very important personalities in mathematics and their work on the concept of Infinity (actually various magnitudes of infinities, I guess), the Continuum Hypothesis, and the Axiom of Choice. While the author takes us back to the ancient Pythagoreans and their determination to keep irrational numbers secret knowledge, the story really centers around Georg Cantor and his struggles in founding the study of mathematics in this field. Cantor was a mystic as well and there is also more than one appearance of the Kabbalah.

Certainly, you can't learn the subject from this book. However, like visiting some vast architectural wonder that you can only take in as a big view, this book places lots of Post It notes on important points if you want to begin reading more deeply about these profound ideas. And if you don't, it is certainly a fund way to spend a few hours.

The author provides four pages of references for further reading, but if, like me, you don't know the field you will likely have to do preliminary studies to just get to the foothills of really taking on the subjects studied in this book. If you already understand the math then this book is likely too light for you unless you somehow missed out on the history of your field.

I enjoyed the book and if you are interested in how serious thinkers learned to think about Infinity and what it actually means, then this book is a fine initial guide.

Interesting book, but not necessary
Aczel wrote this book in layman terms and he did a good job explaining the concept of transfinite numbers.
However, the association of the concept of infinity with another concept, God, is totally nonsense. (Same with the application of the Incompleteness Theorem to god.)

Complete Nonsense
Although the book's content about mathematics is informative, its association of abstract mathematics with Judaism is completely ridiculous. The author tries hard to establish a link between the ponderings of some ancient Rabbis and Cantor's astonishing accomplishments and between some religious concepts and mathematical infinity. Please do not read this book if you are in any way serious about mathematics or science. Over and over again, the author promotes god and the Jewish religion whereas I was just interested in the exciting mathematics of the period. I consider this book propaganda for Judaism and mysticism. It is one of those instances where a religious zealot tries to cast his absurd views into science.

On the inifinity that comes next...
I would never have imagined that a quest this abstract could entail a story so human. Yet, that is exactly what Amir Aczel provides in this smooth tale of the many humbling encounters with the realm of infinity.

There are two lessons from this compact survey on the effort expended and the toll imposed on those bold enough to go where no person can go. First, the urge to comprehend infinity is an ancient quest and inextricably tied to the effort to ascertain the nature of God. Second, getting to know infinity can be massively bad for one's mental health. Mr. Aczel manages an almost impossible task (infinity tends to do that) in this text. He is (a) attempting to survey an enormous amount of the history mathematics and, to some extent, religion, and (b) providing a glimpse into the lives of those mathematicians that have ventured into this field. At the heart of this book is Georg Cantor, founder of modern set theory. Cantor sought to transcent an intuitive understanding of infinity. He sought an ordered system; specifically he sought to prove what became known as the continuim hypothesis: basically, that the lowest order of infinity (some cardinal numbere) was followed by the cardinal number, c (thus permitting Cantor to give ordere to his transfinite numbers). Against this hypothesis stood the possibility, urged by any number of Cantor's opponents, of infininty somewhere before one reached c. The search to prove what Godel later demonstrated to be an undecidable hypothesis may well have led Cantor (and Godel for that matter) to madness. At minimum it may have activated any underlying predisposition to mental illness in both men. They were not, as Aczels's discussion of the Kabbalists shows, inifinity's first victims. Aczel has provided a balanced and very human exploration into a topic that draws its victims as a moth to the flames. ... Read more

Book 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

Reviews (5)

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!

Interesting
I would recommend this book to anyone interested in Mathematics. The fallacies are interesting, including the author's. For example, on page 94 regarding Oscar Wilde's epigram : "The only way to get rid of temptation is to yield to it". Mr. Bunch suggests this to be a fallacy due to the key word "only", and offers an example such as suicide to show "only" to be invalid. But would not suicide be a temptation as well? Or for that manner, anything one would try?

Intriguing
Being a physicist who has always had an interest in math, I've found this book a good way to learn about aspects of mathematics usually saved for metamathematical philosophy courses. The book is not a textbook but it has enough depth for you to understand why paradoxes like Godel's, Grelling's and Russell arise and what their implications are. Each chapter gets more interesting and more complicated than the previous. Even though the book does not have a completely polished feel, I recommend it to any recreational mathematician.

jovial
My 8 year old niece loves her schoolteacher but as she has a bit of devilment in her ... yesterday she ask the school m'arm, "what is 1 divided by 1 - 1"? And where did the little sweetheart get that lovely bit of nothingness? Seems her uncle was reading this book ... : Hey, this is a swell book fun to read with calculator at hand and niece [with her dog ] at knee. ... Read more

Book Description

Here is an excellent undergraduate level text on set theory written in a lively, interesting and good-humored style. This book corresponds to a view of the subject from someone who has thought deeply about this and many other aspects of mathematical logic. The second edition has been expanded to include solutions to the problems, increasing the book's usefulness as a teaching tool. ... Read more