Naive Set Theory

1. Front Matter

   Pages i-vii

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2. The Axiom of Extension

   • Paul R. Halmos

   Pages 1-3

3. The Axiom of Specification

   • Paul R. Halmos

   Pages 4-7

4. Unordered Pairs

   • Paul R. Halmos

   Pages 8-11

5. Unions and Intersections

   • Paul R. Halmos

   Pages 12-16

6. Complements and Powers

   • Paul R. Halmos

   Pages 17-21

7. Ordered Pairs

   • Paul R. Halmos

   Pages 22-25

8. Relations

   • Paul R. Halmos

   Pages 26-29

9. Functions

   • Paul R. Halmos

   Pages 30-33

10. Families

    • Paul R. Halmos

    Pages 34-37

11. Inverses and Composites

    • Paul R. Halmos

    Pages 38-41

12. Numbers

    • Paul R. Halmos

    Pages 42-45

13. The Peano Axioms

    • Paul R. Halmos

    Pages 46-49

14. Arithmetic

    • Paul R. Halmos

    Pages 50-53

15. Order

    • Paul R. Halmos

    Pages 54-58

16. The Axiom of Choice

    • Paul R. Halmos

    Pages 59-61

17. Zorn’s Lemma

    • Paul R. Halmos

    Pages 62-65

18. Well Ordering

    • Paul R. Halmos

    Pages 66-69

19. Transfinite Recursion

    • Paul R. Halmos

    Pages 70-73

20. Ordinal Numbers

    • Paul R. Halmos

    Pages 74-77

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 student 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. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdom is still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.

• addition
• arithmetic
• Cardinal number
• Countable set
• Lemma
• Peano axioms
• set theory

From the reviews:

“This book is a very specialized but broadly useful introduction to set theory. It is aimed at ‘the beginning student of advanced mathematics’ … 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. … A good reference for how set theory is used in other parts of mathematics … .” (Allen Stenger, The Mathematical Association of America, September, 2011)

• Department of Mathematics, Santa Clara University, Santa Clara, USA

  Paul R. Halmos

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