Book Description

This is a softcover reprint of the English translation of 1968 of N. Bourbaki's, Théorie des Ensembles (1970).

Customer Reviews (4)

A Comparison
It seems like theres enough information here (and elsewhere) about the Bourbaki series in general so I wont comment on that. This review is written for someone that wants to learn set theory and is written with an eye to the other options available.

The Bad:

1) No Comparisons - This book constructs one system of axioms for set theory. There is no comparison of this theory with other theories, nor is there any in depth analysis of the axioms and thier interrealtion. You would find this stuff in most other introductory set theory books, but not here.

2) Light Exposition - Set theory and mathematical logic are subjects where results are subtle, and you need to tease out these subtleties by constantly thinking about the theorems. This book, unlike other set theory books, doesnt point out these subtle things and so the burden is left to the reader. (It is a huge burden for the unintiated!)

3) Typographical Errors - There are a few typographical errors. Some of these are obvious, some are not; the ones that are not obvious are devastating, since a proof will become incomprehensible with a few misplaced symbols.

4) Proofs Are Condensed - The proofs offered are written very concisely. There is very little in the proof to help the reader follow along and as such you need to think a bit about what is going on.

The Good:

1) Great System - The particular system of axioms used is rather standard so it wont be unfamiliar. What is especially great is that they go so in depth with the background logic; they actually use part of the logical system and the axioms of set theory to prove the axiom of choice.

2) Rigour - The book is incredibly rigorous. Everything is done with great precision without being long winded.

3) Generality - Theorems are proved in the most general form possible. This is incredibly good if you are interested in using the theorems in many other branches of mathematics. Its also interesting to see exactly how general theorems can get.

4) Structures - This book, unlike other set theory books, focuses heavily on creating mathematical structures from set theoretical objects. If you plan to persue model theory then this is a godsend.

I would not recommend this book to someone who has not already had an aquiantance with axiomatic set theory. I would not recommend this book to anyone that likes to solve mathematical problems: the focus here is on theory and not problem solving. Finally i would not recommend this book to anyone that has even the slightest trouble following rigorous mathematical proofs.

This is a unique book that should be read by good mathematicians after an introduction to set theory; especially if they plan to persue model theory or (obviously) the rest of the Bourbaki series.

Reading Axiomatic Set Theory before this book is more than adequate. Reading Set Theory and Its Philosophy: A Critical Introduction before or afterwards will make up for the shortcomings of this book.

A Wonderful Book
Today, I have just finished reading the first chapter of this amazing book, and I am definitely looking forward to see the rest of it. If you like rigorous mathematics and feel uncomfortable with careless expositions, then Bourbaki is the right choice.

Elements of Mathematics. Theory of Sets
I am neither qualified nor competent to rate any work of "Nicolas Bourbaki"; I will "borrow" from "Wikipedia"about "Nicolas Bourbaki" the following information:
"Nicolas Bourbaki is the collective pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way.

While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki ("association of collaborators of Nicolas Bourbaki"), which has an office at the cole Normale Suprieure in Paris."

Euclid in the XX century
Some historians of Mathematics believe that Euclid, in his great treatise (the Elements) joined together in a consistent way all known Mathematics of his time. Polishing the work of his predecessors, he made important contributions of his own. More than two thousand years later Bourbaki suceeded in making a similar thing with the Mathematics of his time. After the important contributions of the XIX century mathematicians in the foundations of Analysis and of the XX century in Logic, a new way of working with Mathematics emerged. Mathematics did not deal anymore just with numbers, or geometrical figures, or whatever, but with a general and abstract concept called Structure. Much of the development and spreading of this concept is due to the monumental treatise of Bourbaki - the treatise not incidentally is called the Elements of Mathematics. Rewriting all known Mathematics in a very peculiar way, and searching perfect and absolute rigour, Bourbaki perhaps reinvented Mathematics. And it is indisputable its influence in the basic trends of this area of human enterprise. For Bourbaki, the basis of Mathematics is Theory of Sets (including Mathematical Logic). Resting upon it we find Algebra and the Algebraic Structures. Above Algebra is Topology, where we finally meet real numbers. It is said that Bourbaki only wrote about subjects that he considered terminated in a certain way, giving it a final perfected form, but for myself I prefer to think that his intention was similar to Euclid: organize the knowledge. Theory of Sets is the first book of the treatise, that counts ten books to this date and provides the safe foundation on which the whole stuff rests. But it can - and probably must - be read independently of the mysticism involving the treatise, and in my opinion is the best book ever written on the subject, showing what it is all about. Today the Theory of Sets is evolving in unforeseen directions, and of course those directions are not present in Bourbaki's work (search for Thomas Jech's book Set Theory if you want to know what they are doing right now). But the real intention of the classical Theory of Sets, its real purpose and meaning, is no better explained - and I doubt that it can be better explained - than by Bourbaki. ... Read more

Product Description
The name Bourbaki is known to every mathematician. Many also know something of the origins of Bourbaki, yet few know the full story. In 1935, a small group of young mathematicians in France decided to write a fundamental treatise on analysis to replace the standard texts of the time. They ended up writing the most influential and sweeping mathematical treatise of the twentieth century, Les élements de mathématique.Maurice Mashaal lifts the veil from this secret society, showing us how heated debates, schoolboy humor, and the devotion and hard work of the members produced the ten books that took them over sixty years to write. The book has many first-hand accounts of the origins of Bourbaki, their meetings, their seminars, and the members themselves. He also discusses the lasting influence that Bourbaki has had on mathematics, through both the Élements and the Seminaires. The book is illustrated with numerous remarkable photographs.ReadershipStudents, mathematicians, and historians interested in the group of mathematicians known as Bourbaki. ... Read more

Customer Reviews (1)

Interesting for fans of Bourbaki's texts
This is very interesting for fans of any of Bourbaki's mathematics texts.It has photos of many of the members taken at their meetings, and information about how the group operated.You'd never know it from the final product, but their original goal was to write a calculus book!Several of the founding members had just begun teaching, and they were unhappy with the standard French calculus text of the day--a multivolume work by Goursat that they considered out of date.They decided to write a little bit of background material on algebra and general topology, and somehow ended up with the books we all know and love.
The book also describes the personality quirks of the members, and has some commentary on the contents of the various texts of Bourbaki.It doesn't explain technical details like why Bourbaki chose to define integrals the way they did.
If you didn't major in math, this book probably won't interest you. ... Read more

Book Description
This is the softcover reprint of the English translation of 1975 (available from Springer since 1989) of the first 3 chapters of Bourbaki's 'Groupes et algèbres de Lie'. The first chapter describes the theory of Lie algebras, their derivations, their representations and their enveloping algebras. In Ch. 2, free Lie algebras are introduced in order to discuss the exponential, logarithmic and the Hausdorff series. Ch. 3 deals with the theory of Lie groups over R and C and ultrametric fields. It describes the connections between their local and global properties, and the properties of their Lie algebras. It is one of the very best references on this subject. ... Read more

Customer Reviews (1)

great as references, and..
...even better when supplemented by other texts (be wary though, as the notation of Bourbaki is not universally accepted).
I'd buy themsimply for their sparkling clarity and careful attention to rigour (esp. Commalg, Lie theory and integration).
These arewonderfully crafted, masterful expositions, and shouldnot, by any means, be overlooked by the (aspiring) pure mathematician.

Cheers,
A. ... Read more

Book Description

Intégration is the sixth and last of the Books that form the core of the Bourbaki series; it draws abundantly on the preceding five Books, especially General Topology andTopological Vector Spaces, making it a culmination of the core six. The power of the tool thus fashioned is strikingly displayed in Chapter II of the author's Théories Spectrales, an exposition, in a mere 38 pages, of abstract harmonic analysis and the structure of locally compact abelian groups.

Book Description
This is the softcover reprint of the English translation of 1974 (available from Springer since 1989) of the first 3 chapters of Bourbaki's 'Algèbre'. It gives a thorough exposition of the fundamentals of general, linear and multilinear algebra. The first chapter introduces the basic objects: groups, actions, rings, fields. The second chapter studies the properties of modules and linear maps, especially with respect to the tensor product and duality constructions. The third chapter investigates algebras, in particular tensor algebras. Determinants, norms, traces and derivations are also studied. ... Read more

Book Description
This is the softcover reprint of the English translation of 1971 (available from Springer since 1989) of the first 4 chapters of Bourbaki's Topologie générale. It gives all the basics of the subject, starting from definitions. Important classes of topological spaces are studied, uniform structures are introduced and applied to topological groups. Real numbers are constructed and their properties established. Part II, comprising the later chapters, Ch. 5-10, is also available in English in softcover. ... Read more

Customer Reviews (2)

a solid introduction
this book is a bit denser than most other introductory general topology books. But it does quite exhaustive survey of important concepts pertaining to general topology.

Since Bourbaki series builds upon its previous materials, many set theoretic ideas and terminologies are used without explanations. So unless one does have access to their previous book "Theory of Sets" there will be some minor frustrations/annoyances when reading this book.

the book that the experts study
i recently ordered "A GENERAL TOPOLOGY WORKBOOK" by Iain Adamson, as i very much wish to understand the mathematics of topology.as i would do with any other book, i started by reading the introduction and suggested readings.Mr. Adamson highly recommends the BOURBAKI series on topology as a reference material.he ascribes to this series as his text of choice and further states that this is the text that he has studied the most closely.it is for this reason that i am ordering this series.

i extend my thanks to mr. adamson for the recommendation.with the plethora of choices in study materials and a limited budget, i needed to narrow my scope and decide which text would best serve me.

thanks again:-) ... Read more

Book Description
This is the softcover reprint of the English translation of 1972 (available from Springer since 1989) of the first 7 chapters of Bourbaki's 'Algèbre commutative'. It provides a very complete treatment of commutative algebra, enabling the reader to go further and study algebraic or arithmetic geometry. The first 3 chapters treat in succession the concepts of flatness, localization and completions (in the general setting of graduations and filtrations). Chapter 4 studies associated prime ideals and the primary decomposition. Chapter 5 deals with integers, integral closures and finitely generated algebras over a field (including the Nullstellensatz). Chapter 6 studies valuation (of any rank), and the last chapter focuses on divisors (Krull, Dedekind, or factorial domains) with a final section on modules over integrally closed Noetherian domains, not usually found in textbooks. Useful exercises appear at the ends of the chapters. ... Read more

Book Description
From the reviews of the French edition: "This is a rich and useful volume. The material it treats has relevance well beyond the theory of Lie groups and algebras, ranging from the geometry of regular polytopes and paving problems to current work on finite simple groups having a (B,N)-pair structure, or "Tits systems". A historical note provides a survey of the contexts in which groups generated by reflections have arisen. A brief introduction includes almost the only other mention of Lie groups and algebras to be found in the volume. Thus the presentation here is really quite independent of Lie theory. The choice of such an approach makes for an elegant, self-contained treatment of some highly interesting mathematics, which can be read with profit and with relative ease by a very wide circle of readers (and with delight by many, if the reviewer is at all representative)."
(G.B. Seligman in MathReviews) ... Read more

Product Description

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Ce volume du Livre d Algèbre commutative, septième Livre du traité, comprend les chapitres: 1. Dimension; 2. Anneaux locaux noethériens complets.

Le chapitre 8 traite de diverses notions de dimension en algèbre commutative, telles que la dimension de Krull d un anneau. Ces notions jouent un rôle capital en géometrie algébrique. Le chapitre 9 introduit, quant à lui, les vecteurs de Witt et les anneaux japonais.

Ce volume est une réimpression de l édition de 1983.

Book Description
This is the softcover reprint of the English translation of 1974 (available from Springer since 1989) of the later chapters of Bourbaki's Topologie générale. It completes the treatment of general topology begun in Part I (Ch. 1-4, also available in English in softcover). The real numbers having been introduced in Ch. 4, the first chapters of this volume study subgroups and quotients of R (with applications to the 'measurement of magnitudes' and to the log and exp functions), then real vector spaces and projective spaces, then the additive groups Rn (subgroups, quotients, homomorphisms, infinite sums and products). Analogous properties are then studied for complex numbers, in Ch.8. Chapter 9 illustrates the use of real numbers in general topology, studying different important kinds of topological spaces: uniformizable, metric, normal Baire, Polish, Borel spaces.The final chapter deals with the various topologies of function spaces,ending with a section on approximation of functions. ... Read more

Customer Reviews (1)

Djerassi did not win a nobel prize.
This excellent book explored the collaborative process of science in the framework of an interesting story. It is a must read for scientists who should learn lessons from this book.It should be noted however that theauthor is not a nobel prize winner as the first review would have usbelieve. ... Read more

Book Description

Integration is the sixth and last of the books that form the core of the Bourbaki series; it draws abundantly on the preceding five Books, especially General Topology andTopological Vector Spaces, making it a culmination of the core six. The power of the tool thus fashioned is strikingly displayed in Chapter II of the author's Théories Spectrales, an exposition, in a mere 38 pages, of abstract harmonic analysis and the structure of locally compact abelian groups.

Book Description

This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981).

Book Description

This is the English translation of Bourbaki's text Groupes et Algèbres de Lie, Chapters 7 to 9. It completes the previously published translations of Chapters 1 to 3 (3-540-64242-0) and 4 to 6 (3-540-42650-7) by covering the structure and representation theory of semi-simple Lie algebras and compact Lie groups. Chapter 7 deals with Cartan subalgebras of Lie algebras, regular elements and conjugacy theorems. Chapter 8 begins with the structure of split semi-simple Lie algebras and their root systems. It goes on to describe the finite-dimensional modules for such algebras, including the character formula of Hermann Weyl. It concludes with the theory of Chevalley orders. Chapter 9 is devoted to the theory of compact Lie groups, beginning with a discussion of their maximal tori, root systems and Weyl groups. It goes on to describe the representation theory of compact Lie groups, including the application of integration to establish Weyl's formula in this context. The chapter concludes with a discussion of the actions of compact Lie groups on manifolds. The nine chapters together form the most comprehensive text available on the theory of Lie groups and Lie algebras.

Book Description

This book is an English translation of the last French edition of Bourbakis Fonctions d'une Variable Réelle.

Book Description
This is a softcover reprint of the English translation of 1987 of the second edition of Bourbaki's Espaces Vectoriels Topologiques (1981).
This [second edition] is a brand new book and completely supersedes the original version of nearly 30 years ago. But a lot of the material has been rearranged, rewritten, or replaced by a more up-to-date exposition, and a good deal of new material has been incorporated in this book, all reflecting the progress made in the field during the last three decades.
Table of Contents.
Chapter I: Topological vector spaces over a valued field.
Chapter II: Convex sets and locally convex spaces.
Chapter III: Spaces of continuous linear mappings.
Chapter IV: Duality in topological vector spaces.
Chapter V: Hilbert spaces (elementary theory).
... Read more

Book Description

Lobjet de cet ouvrage est une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Product Description

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Ce volume du Livre d Algèbre commutative, septième Livre du traité, est la continuation des chapitres antérieurs. Il introduit notamment les notions de profondeur et de lissité, fondamentales en géometrie algébrique. Il se termine par l introduction des modules dualisants et de la dualité de Grothendieck.

Ce volume est paru en 1998.