Reviews of Naive set theory by Paul R Halmos

Paul Halmos wrote Naive set theory which is owned by a remarkable number of mathematicians who, like me [EFR] studied in the 1960s. Because this book seems to have received such a large number of reviews we devote a separate paper to this book.

For extracts from reviews and Prefaces of other books by Halmos we have split our collection into two parts. For the first part of the collection, see THIS LINK. For the second part of the collection, see THIS LINK.

Naive set theory (1960), by Paul R Halmos.
1. From The Preface.

Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning students of advanced mathematics the basic set-theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. . . . Instead of Naive Set Theory a more honest title for the book would have been An Outline of the Elements of Naive Set Theory. "Elements" would warn the reader that not everything is here; 'outline" would warn him that even what is here needs filling in ... . The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it... the language and notation are those of ordinary informal mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study. ... By way of examples we might occasionally speak of sets of cabbages, and kings, and the like, but such usage is always to be construed as an illuminating parable only, and not as a part of the theory that is being developed.

2. Review by: Elliott Mendelson.
The Journal of Philosophy 57 (15) (1960), 512-513.

Those of us who have been so pleasantly introduced into the intricacies of linear algebra and measure theory by Paul Halmos will not be disappointed by his new excursion into the realm of set theory. Here is a book on mathematics which does not have the repelling format of lists of definitions, theorems, and proofs. It is "naive" only in the sense that it is informal, contains a mini- mum of special symbols, and has a charming conversational style. ... Although the book is intended to tell the "basic set-theoretic facts of life " to beginning students of advanced mathematics, it can be read with pleasure by all, and with great profit by those willing to fill in the details of the proofs.

3. Review by: Harry M Gehman.
Philosophy and Phenomenological Research 22 (1) (1961), 122-123.

Halmos writes as a mathematician. He uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know. Although axioms for set theory are stated and are used as a basis for proofs, there is a minimum of logical formalism. Usually a theorem is followed by a sketch of a proof rather than by a formal deduction. Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics.

4. Review by: L Rieger.
Mathematical Reviews MR0114756 (22 #5575).

First of all, the word "naive", in the title of this very concise and lucid introduction to set theory, disagrees with its common meaning (naive set theory=the preaxiomatic, Cantorian, absolute one). Perhaps the adjective "naive" is due to the author's modesty; in fact, the text-book is built on a didactically excellent, informal but exact exposition of the Zermelo-Fraenkel-Skolem axioms and of their standard consequences ("...axiomatic set theory from the naive point of view'', as the author says in the preface). In an almost conversational though not uncritical style, the author quickly proceeds from basic facts to more involved ones, by means of a great variety of instructive examples. If one disregards problems of the foundations and needs only "some'' set theory (as a basic standard tool only), this seems to be the best mode. ... the reviewer wants to say that the book is to be recommended as one of the best modern introductory text books of set theory.

5. Review by: J Richard Buchi.
Philosophy of Science 28 (4) (1961), 445.

This book is intended to be an introduction to set theory for pregraduate students of mathematics. It provides a survey of how the basic facts of the arithmetic of natural numbers, ordinals, and cardinals may be derived from a standard system of axioms on the relation of set-membership. The approach is "semi-axiomatic", in the sense that decision as to what constitutes a proof from the axioms is left to intuition (presumably the author's, or a student's trained in other mathematics courses). Nevertheless, at some crucial places (Aussonderungsaxiom), it is hinted that elementary quantification theory might be introduced to make these matters more precise. The reviewer would have liked to see a somewhat less naive discussion of the "naive approach" to set theory. Appeal is made to geometry. There, however, the situation is much more clear because one starts from an admittedly vague but rather intricate "naive model".

6. Review by: A Heyting.
Synthese 13 (1) (1961), 86-87.

Halmos writes for mathematicians, who have to know the principal things of it, without making the set theory their special subject of study. His way through the matter is a straight one ; starting from the axiomas he deduces the principal propositions. Treated thus, the set theory appears as a series of facts about a reality outside us, rather like euclidian geometry, founded on axiomas, was conceived as a series of facts about the space surrounding us. This conception of mathematics as concerning an outward knowledge of facts is considered as naive by the modern mathematician. Halmos' book has been clearly written and is very suitable to learn the principles of the set theory from it. The argumentation is, though complete, briefly formulated, so that the reader who is not a skilled mathematician, will find it difficult in some places.

7. Review by: Sidney G Winter, Jr.
Journal of the American Statistical Association 56 (296) (1961), 1022-1023.

Little attention is paid to the axioms, per se - to such questions as consistency, independence, possible alternative formulations, etc. There is barely a hint, for example, that the axiom of choice is in any sense more controversial than the other axioms introduced. The emphasis is on a rapid development of concepts and results rather than on a thorough examination of each step that is taken. In spite of the almost total absence of examples and the fast pace (with many steps left for the reader to verify), the presentation is lucid.

8. Review by: R L Goodstein.
The Mathematical Gazette 45 (354) (1961), 375.

There is nothing naive about the first of these books; the adjective is used in the title only to stress the absence of formal-logic notation. Halmos' account is[very]readable but is marred by the author's view that set theory is a trifle to be absorbed swiftly and forgotten.

9. Review by: Alfons Borgers.
The Journal of Symbolic Logic 34 (2) (1969), 308.

It is the purpose of this purely expository book to tell the beginning student of advanced mathematics the basic principles and facts of set theory. The treatment is called naive because the language used is the same as that of ordinary informal mathematics. But as a matter of fact the treatment is axiomatic because the proofs are based on the axioms which are commonly used in the well-known textbooks. A comparative study of alternative sets of axioms is not contemplated. ... The style is almost conversational and makes for pleasant reading. It has the drawback that some proofs are not as complete as might be desirable. Only the essential theorems are displayed and the reader should unearth the other ones from the informal explanations.

10. Review by: H Mirkil.
Amer. Math. Monthly 68 (4) (1961), 392.

Having worked both in formal logic and in plain mathematics, Halmos is admirably qualified to write this book. It is possible that he was first prompted to set pen to paper when (as a Van Nostrand editor) he saw the manuscript of Suppes's Axiomatic Set Theory. In any event the two books form a natural contrasting pair, Suppes's for the careful logician investigating (say) the independence of the axiom of choice, Halmos's for the mathematician-in-the-street who just wants to stay out of trouble when he does (say) measure theory. One small cavil. It seems to the reviewer that Halmos should have relaxed his "naive" principles occasionally, at least in his examples. A mathematician indifferent to most logical subtleties might nonetheless be delighted by the weird un-integer-like objects that happen to satisfy Peano's postulates.

JOC/EFR August 2016

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