Ebook: An Outline of Set Theory

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.

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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. AN OUTLINE OF SET THOERY is organized into three parts: the first contains definitions and statements of problems, the second contains suggestions for their solution, and the third contains complete solutions.

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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. AN OUTLINE OF SET THOERY is organized into three parts: the first contains definitions and statements of problems, the second contains suggestions for their solution, and the third contains complete solutions.
Content:
Front Matter....Pages i-viii
Introduction....Pages 1-4
Front Matter....Pages 5-5
Logic and Set Theory....Pages 7-13
The Natural Numbers....Pages 15-19
The Integers....Pages 21-22
The Rationals....Pages 23-24
The Real Numbers....Pages 25-26
The Ordinals....Pages 27-32
The Cardinals....Pages 33-36
The Universe....Pages 37-40
Choice and Infinitesimals....Pages 41-44
Goodstein’s Theorem....Pages 45-48
Front Matter....Pages 49-49
Logic and Set Theory....Pages 51-53
The Natural Numbers....Pages 55-56
The Integers....Pages 57-58
The Rationals....Pages 59-61
The Real Numbers....Pages 63-66
The Ordinals....Pages 67-70
The Cardinals....Pages 71-79
The Universe....Pages 81-83
Choice and Infinitesimals....Pages 85-89
Front Matter....Pages 49-49
Goodstein’s Theorem....Pages 91-93
Front Matter....Pages 95-95
The Integers....Pages 97-99
The Rationals....Pages 101-104
The Real Numbers....Pages 105-108
The Ordinals....Pages 109-114
The Cardinals....Pages 115-117
The Universe....Pages 119-121
Choice and Infinitesimals....Pages 123-127
Goodstein’s Theorem....Pages 129-132
Back Matter....Pages 133-135
....Pages 137-139