An Outline of Set Theory

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   1. Introduction

      • James M. Henle

      Pages 1-4

3. Projects

   1. Front Matter

      Pages 5-5

      PDF

   2. Logic and Set Theory

      • James M. Henle

      Pages 7-13

   3. The Natural Numbers

      • James M. Henle

      Pages 15-19

   4. The Integers

      • James M. Henle

      Pages 21-22

   5. The Rationals

      • James M. Henle

      Pages 23-24

   6. The Real Numbers

      • James M. Henle

      Pages 25-26

   7. The Ordinals

      • James M. Henle

      Pages 27-32

   8. The Cardinals

      • James M. Henle

      Pages 33-36

   9. The Universe

      • James M. Henle

      Pages 37-40

   10. Choice and Infinitesimals

       • James M. Henle

       Pages 41-44

   11. Goodstein’s Theorem

       • James M. Henle

       Pages 45-48

4. Suggestions

   1. Front Matter

      Pages 49-49

      PDF

   2. Logic and Set Theory

      • James M. Henle

      Pages 51-53

   3. The Natural Numbers

      • James M. Henle

      Pages 55-56

   4. The Integers

      • James M. Henle

      Pages 57-58

   5. The Rationals

      • James M. Henle

      Pages 59-61

   6. The Real Numbers

      • James M. Henle

      Pages 63-66

   7. The Ordinals

      • James M. Henle

      Pages 67-70

This book is designed for use in a one semester problem-oriented course in undergraduate set theory. The combination of level and format is somewhat unusual and deserves an explanation. Normally, problem courses are offered to graduate students or selected undergraduates. I have found, however, that the experience is equally valuable to ordinary mathematics majors. I use a recent modification of R. L. Moore's famous method developed in recent years by D. W. Cohen [1]. Briefly, in this new approach, projects are assigned to groups of students each week. With all the necessary assistance from the instructor, the groups complete their projects, carefully write a short paper for their classmates, and then, in the single weekly class meeting, lecture on their results. While the em­ phasis is on the student, the instructor is available at every stage to assure success in the research, to explain and critique mathematical prose, and to coach the groups in clear mathematical presentation. The subject matter of set theory is peculiarly appropriate to this style of course. For much of the book the objects of study are familiar and while the theorems are significant and often deep, it is the methods and ideas that are most important. The necessity of rea­ soning about numbers and sets forces students to come to grips with the nature of proof, logic, and mathematics. In their research they experience the same dilemmas and uncertainties that faced the pio­ neers.

• Finite
• calculus
• cardinals
• mathematics
• ordinal
• set theory
• theorem

• Department of Mathematics, Smith College, Northampton, USA

  James M. Henle

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