This book is an innovative problem-oriented introduction to undergraduate set theory. It is intended to be used in a course in which the students work in groups on projects and present their solutions to the class. Students completing such a course come away with a deeper understanding of the material, as well as a clearer view of what it means to do mathematics. The topics covered include standard undergraduate set theory, as well as some material on nonstandard analysis, large cardinals, and Goodstein's Theorem.

One Projects.- 1. Logic and Set Theory.- 2. The Natural Numbers.- 3. The Integers.- 4. The Rationals.- 5. The Real Numbers.- 6. The Ordinals.- 7. The Cardinals.- 8. The Universe.- 9. Choice and Infinitesimals.- 10. Goodstein's Theorem.- Two Suggestions.- 1. Logic and Set Theory.- 2. The Natural Numbers.- 3. The Integers.- 4. The Rationals.- 5. The Real Numbers.- 6. The Ordinals.- 7. The Cardinals.- 8. The Universe.- 9. Choice and Infinitesimals.- 10. Goodstein's Theorem.- Three Solutions.- 1. Logic and Set Theory.- 2. The Natural Numbers.- 3. The Integers.- 4. The Rationals.- 5. The Real Numbers.- 6. The Ordinals.- 7. The Cardinals.- 8. The Universe.- 9. Choice and Infinitesimals.- 10. Goodstein's Theorem.

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.