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Naïve Set Theory

Paul R. Halmos
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
BLL Rating: 

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This book is a very specialized but broadly useful introduction to set theory. It is aimed at “the beginning student of advanced mathematics” (p. v) who wants to understand the set-theoretic underpinnings of the mathematics he already knows or will learn soon. It is also useful to the professional mathematician who knew these underpinnings at one time but has now forgotten exactly how they go.

The “Naïve” in the title does not mean “For Dummies”, but is used in contrast to “Axiomatic”. The book does present Zermelo-Fraenkel set theory, and shows two or three axioms explicitly, but it is not an axiomatic development. The present work is a 1974 reprint of the 1960 Van Nostrand edition, and so just missed Cohen’s 1963 resolution of the continuum hypothesis.

The book does not intend to present a comprehensive treatment of set theory. It is something of a hodge-podge, without a clear narrative thread, and choppy at times. It covers simple things such as functions and unordered and ordered pairs, less-simple things such as indexing by general sets, and intricate things such as transfinite induction and recursion. It gives two explanations of numbers, first through the Peano postulates and then a fairly thorough development of finite and transfinite ordinals and cardinals following Cantor. There are occasional exercises scattered through the book, but nothing systematic. Van Nostrand published a companion exercise book in 1966 (Exercises in Set Theory by L. E. Sigler, reprinted by Springer in 1976), but this is out-of-print and I have not seen a copy.

Bottom line: A good reference for how set theory is used in other parts of mathematics, but not a text or reference on set theory per se.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

Preface. 1: The Axiom of Extension. 2: The Axiom of Specification. 3: Unordered Pairs. 4: Unions and Intersections. 5: Complements and Powers. 6: Ordered Pairs. 7: Relations. 8: Functions. 9: Families. 10: Inverses and Composites. 11: Numbers. 12: The Peano Axioms. 13: Arithmetic. 14: Order. 15: The Axiom of Choice. 16: Zorn's Lemma. 17: Well Ordering. 18: Transfinite Recursion. 19: Ordinal Numbers. 2: Sets of Ordinal Numbers. 21: Ordinal Arithmetic. 22: The Schr”der-Bernstein Theorem. 23: Countable Sets. 24: Cardinal Arithmetic. 25: Carnidal numbers.