Set theory and logic [antikvár]

PREFACE This book is an outgrowth of a course which I have developed at Oberlin College for advanced undergraduates. The purpose of the course is to introduce students to the foundations of mathematics, to give them initial training in the axiomatic method in mathematics, and to provide them with the necessary tools to cope successfully with graduate level courses having an abstract and axiomatic orientation. It may be inferred that as I use the term "foundations of mathematics" I understand it to mean an analysis of fundamental concepts of mathematics, intended to serve as a preparation for studying the superstructure from a general and unified perspective. The book contains adequate material for any one of a variety of one-year upper undergraduate courses with a title resembling "Introduction to Foundations of Mathematics." That is, there is sufficient material for a year's course in which the instructor chooses to emphasize the construction of standard mathematical systems, or the role of logic in connection with axiomatic theories, or, simply, mathematical logic. Further, by focusing attention on certain chapters, it can serve as a text for one-semester courses in set theory (Chapters 1, 2, 5, 7), in logic (Chapters 1, 4, 5, 6, 9), and in the development of the real number system (Chapters 1, 2, 3, 5, 8). The book has been organized so that not until the last chapter does symbolic logic play a significant role. Most of the material presented might be described as the mathematics whose development was directly stimulated by investigations pertaining to the real number system. That is, the development and the study of the real number system serve as the underlying theme of the book. I will elaborate on this statement after outlining the contents. Chapter 1 is an introduction to so-called intuitive set theory. Along with the algebra of sets the theory is developed to the point where the notion of a relation can be defined. The remainder of the chapter is concerned with the special types of relations called equivalence relations, functions, and ordering relations. Sufficient examples and exercises are provided to enable the beginner to assimilate these concepts fully. Chapter 2 begins with a discussion of a type of system (an "integral system") which incorporates several features of the natural number