Customer Reviews

Axiomatic Set Theory (Dover Books on Mathematics)

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Mathematics is a first order theory whose primitive formulae
all take the form 'a is a member of b'. 'a' can be a set or atom; 'b' must be a set. If you do not object to the preceding sentence, then read on.

Axiomatic Set Theory (AST) lays down the axioms of the now-canonical set theory due to Zermelo, Fraenkel (and Skolem), called ZFC. Building on ZFC, Suppes then derives the theory of cardinal and ordinal numbers, the integers, rationals, and reals, and the transfinite--Cantor's paradise. Suppes accomplishes in 250 well laid out pages what required 800 crabbed pages in Principia Mathematica.

This book evolved out of a class Suppes taught at Stanford in the long ago 1950s. It has since remained the best book of its kind. The reason is that subsequent presentations of set theory are too difficult, too contrived, too clever by half. They disdain the basics as old hat.

AST has several valuable pedagogical features.

1. The introduction to relations and functions is the best I know of. I am disappointed at how little attention has been devoted to relations and relational algebra in recent decades.

2. Suppes has a nice way of introducing a simple axiom, then showing that that axiom is a theorem when a more complicated axiom is later introduced. In particular, he develops the theory of cardinals by means of a temporary axiom to the effect that equipollent sets have identical cardinalities. This axiom becomes a theorem when the axiom of Choice is introduced in the final chapter.

The axiom schema of Replacement is introduced as late as possible, to enable transfinite arithmetic. He then turns around and shows that Replacement makes Subsets and Pairing redundant.

In my opinion, the greatest flaw of ZFC is that defining a cardinal number requires either the axiom of Choice, or Infinity plus the subtle notion of set rank. Frege and Russell had an appealing definition: a cardinal number is an equivalence class of sets under equipollence. That definition does not work in ZFC. It does work in Quinian set theory. Suppes does a yeoman's job of battling this flaw.

3. Suppes defines a finite set in the interesting way Tarski proposed in 1924.


AST contains hundreds and hundreds of theorems, man of them useful classics. In many cases, the proof is an exercise. Suppes's proof are of the informal sort typical of mathematics. What AST does can be done more rigorously: type 'Metamath' into Google and see for yourself.

Even though Suppes is a philosopher, this book is almost entirely a mathematical exercise. The reader will not get a good feel for how set theory is part of analytic philosophy, and how it has been a contentious subject. The writings of Fraenkel and Bar Hillel are better in these respects. Suppes does highlight the reservations re the axiom of Choice, but Cohen's proof that Choice is independent of ZF has largely laid those reservations to rest, except for those of us with constructive sympathies. AST gives no hint that Replacement and Power Set give us far more set theory than is needed in practice. Thanks to the work of Aczel and Barwise, published around 1990, we have a better idea of what it means to dispense with Regularity. Shortly after the 1960 publication of AST, Lawvere and others began to lay down the category theoretic foundation of mathematics knowns as topos theory. That theory puts ZFC in a new light. Personally, I am astonished that an axiomatization of finite sets simpler than ZF has yet to emerge.

This book presents a rigorous, axiomatic development of classic set theory, introducing the axioms as needed and founding nearly all results upon theorems derived earlier in the book (or on the axioms themselves). It is genuinely gratifying to see the development proceed in such a regimented fashion, from basic sets to natural numbers to reals, and then on to transfinite induction and the axiom of choice. There are numerous exercises; no answers are provided, but the intelligent reader who proceeds carefully should not find this a hindrance. It is however, not a modern book; readers who want to understand current ideasin set theory (inaccessible, supercompact cardinals, etc.) should look elsewhere.

One does not hear about set theory too much these days, no doubt due to the de-emphasis of foundational discussions in mathematics. Foundational questions of course were the focus of much attention in mathematics in the early twentieth century, this taking place because of the many paradoxes in set theory and due to the influence of the philosophers. Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning. This book could serve to introduce these topics or as an historical reference to the issues as they were hotly debated in the last century.

The first chapter gives an informal introduction to the notion of a set, first-order predicate logic (notions of bound and free variables and quantification), and the Zermelo-Fraenkel axioms of set theory. The author describes the difficulties in the "axiom of abstraction" in the writings of Frege as pointed out by Bertrand Russell. It is pointed out that the axiom of abstraction is in fact an infinite collection of axioms, thus motivating the concept of an "axiom schema". The axiom schema that is used explicitly in the book is the "axiom schema of separation" due to Ernst Zermelo, which he formulated in order to make precise the notion of a statement as being "definite". More of the set-theoretic paradoxes are discussed, along with their classification due to F.P. Ramsey into "linguistic" and "semantical" ones.

The advantage of an older book on set theory is that more of the underlying details are explained, instead of just being formally developed. The author gives a thorough discussion of the concepts throughout the book, beginning with an organized development in chapter 2. He begins immediately with discussing the distinction between the object language and metalanguage, and the symbols to be used in the object language: constants, variables, logical connectives, quantifiers, and grouping symbols. These symbols are used to construct formulas, a subclass of which, the primitive formulas, are defined recursively, and which all formulas in the object language can be expressed in terms of. Throughout the book though the author uses additional notation that allows formulas not to be written in terms of primitive formulas. This is done to make the text more readable, but he requires that the added notion satisfy the criterion of eliminability and non-creativity. The notion of a set is defined formally, and then the axiom of extensionality, which gives a criterion for two sets being equal, and the axiom schema schema of separation. The pairing axiom, which gives the existence of a non-empty set; the sum axiom, which gives the existence of the union of a family of sets; the power set axiom, which gives the notion of the set of all subsets of a set; and the axiom of regularity, which prohibits infinite descending sequences of sets, are all discussed in detail.

Chapter 3 treats relations and functions, so important not only in mathematics but in computer science, especially in the theory of relational databases. Then in chapter 4, the author begins a study of cardinality and the cardinal numbers, proving that the finite cardinal numbers have the properties of the natural numbers, as one would expect. The author is careful to point out the need for the axiom of cardinal numbers in this study. Chapter 5 then goes into the theory of ordinal numbers, wherein it is emphasized that no special axioms are needed for the development of this theory. The author is also careful to note the special problems that arise in defining the arithmetic of natural numbers, such as defining addition recursively without using set theory. But including the apparatus of set theory does allow the replacement of the recursive definition by a proper definition. The axiom of infinity is brought in to permit the construction of arithmetical operations as certain sets. The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals.

The author then shows that set theory can allow the construction of the real numbers, which takes place after the construction of the rational numbers. The famous "Dedekind cut" is discussed, along with the method of Cantor, which defines real numbers as equivalence classes of Cauchy sequences of rational numbers. The author uses the Cantor approach in the rest of the book. He also proves the famous Cantor theorem on the non-denumerability of the real numbers, and gives a brief discussion of the Continuum Hypothesis.

Chapter 8 then gives an overview of the fascinating topics of transfinite induction and ordinal number theory. Recursion theory makes its appearance again in the transfinite recursion for ordinal numbers, using the axiom schema of replacement. The non-commutativity of ordinal addition and multiplication is brought out, and the falsity of Fermat's Last Theorem and Goldbach's Hypothesis in ordinal number theory is shown. The author then shows to what extent cardinal number theory can be done without using special axioms by defining cardinal numbers as initial ordinals. The axiom of choice however is needed to show that every set has a cardinal number. The author then restates the Zermelo-Fraenkel axioms in their final form at the end of the chapter.

The final chapter gives an overview of the most controversial topic in all of set theory, if not in all of mathematics: the axiom of choice. The author shows that the use of this axiom allows one to prove that an infinite set has a denumerable subset, and he shows the equivalence of the axiom of choice with the numeration theorem, the well-ordering theorem, Zorn's lemma, and the law of trichotomy. The counterintuitive Banach-Tarski paradox is discussed, and the author shows the existence of axioms which imply the axiom of choice.

This is a basic introduction to axiomatic set theory. You dont need much experience with informal set theory or formal logic to begin it. The book is rigorous and follows a definition - theorem - proof format, broken with clear exposition and historical notes. Enough formal mathematical logic is introduced only to express the axioms (that is, formal proof systems are not used or discussed). He uses the ZF (Zermelo/Fraenkel) system and gives footnote comparisons to the NBG (von Neumann/Bernays/Godel) system. In chapter 4 he introduces a special axiom (outside of ZF) to simplify the development of cardinal numbers, and this involves the addition of a new primitive notion. All theorems relying on this special axiom are clearly marked. While this admirably allows Suppes to avoid employing the axiom of choice or taking the extra time needed to develop cardinals strictly within ZF, it may annoy some readers. In chapter 6 he gives a detailed construction of the rational numbers and the real numbers (using Cauchy sequences). He uses only the axiom of separation until the axiom of replacement becomes necessary. He does a good job explaining why each axiom is needed and how it arose historically. The book is comparable to Monk's _Introduction to Set Theory_, though a little easier and less advanced.

A set, wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements or members of the set. With such sets, one can do a great many useful manipulations. In the words of Bourbaki: "Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, The Theory of Sets." Set theory, however, rests on a number of axioms -- one might say, circumscriptions. Interestingly, without the formulation of such axioms, the theory potentially runs into serious contradictions.

This book is a classic, and it rigorously, compellingly, and comprehensively introduces each of the axioms of axiomatic set theory, explaining precisely why the axiom is required, and how it came to be developed historically. This it does, however, in a very sparse and dense manner. It would seem to read something like an annual auditor's report. In the author's own words, "illustrative examples are given only sparingly". Thus the first four chapters cover "the foundations of mathematics [for] philosophy courses", while the book as a whole covers "a semester mathematics course in set theory for seniors".

I came to this book with a good grasp of naive set theory and logic. With this, I considered that it would enable me to grasp axiomatic set theory. The author states in the Preface: "No previous knowledge of set theory or mathematical logic is assumed." Almost immediately, however, the book introduced symbols and concepts which, with the information provided, were simply inscrutable. I showed my questions to a B.Sc. undergraduate. He was baffled -- despite having studied set theory. Thus the book would ideally require, I believe, a particularly good background in set theory and logic, or a competent tutor to walk beside one.

This was the first book I have read about set theory, so obviously I cannot compare it to others. It motivates the subject very well and the proofs of the theorems are generally not very hard.
It is unfortunate, that many proofs are not given, but are given as exercises only. This limits its use for self study a bit in my opinion, and this is why I did not give it full five stars.

The book is virtually free of typos! (the book was written 40 years ago, and maybe there was a time, when proof reading was taken more seriously...)

This book is a basic reading on the Set Theory field. It works as an introductory and reference text, with a clear and concise style and a precise logic sequence on its exposition and development of contents. After reading it, I started to focus on more specific topics as Axiom of Choice and Banach-Tarski paradox, and I felt that it has prepared me to go further in my Math studies.

This is an excellent book for an undergraduate who has to begin doing abstract proof and has not had even rudimentary logic and symbolism.

I skim this book one day while looking for some reference books at my local bookstore.The clarity of the book makes it a good book for complementing set theory courses.The examples are given in a consice manner without obstructing the learning material to the readers. However, the lack of answers to the given problems makes the book unfitting to first time readers who may want to learn the subject just for plain curiosity; none of the steps are hinted for solving any problem. It is unfair having to buy this book because readers buying this book are expected to be experts in the field. Rather, it should also consider general science readers who have interest for the subject and want to learn the material. I think having full answers to problems allows all readers to have a good understanding of the subject at hand since it clarifies the bridge of ideas that mathematicians are trying to let the world see. Furthermore, it motivates readers to critically think challenging problems once enough practice has been establish through the ones with answers. However, because there is a lack of communication and guidance to achieve a full census with the subject, this book has a mild sour taste. So in hopes that the author of this book improves the book for the benefit of his readers, "Please provide answers next time."

It's really difficult to share the overwhelming enthousiasm shown in the other reviews : the author's treatment of functions and of recursive definitions is a pure indigestion... One wonders what beautiful book it could have been if written by true didactically-talented authors such as Gauss, Hardy, Apostol, Tarski, Coxeter, Courant, Kline, Greenberg, Russsell (except for the Principia, his crowning example of unreadable masterpiece), Quine, Smullyan...

I'm painfully trying to get to the end of Suppes's cascade of definitions, theorems, definitions... requiring just as much of a cascade of annotations to clarify his text !

On the thin positive side, Suppes' book is a little better than another, highly constipated, book on the same subject i.e; Potter's "Set theory and its philosophy"...