Existence, Truth, and Method In the Mathematical Theory of Sets

1 Preface In these three chapters I treat a variety of issues that surround the current state of set theory. Through an analysis of the relationships between set theory and other branches of mathematics, I try to paint a truer picture of the nature of the theory of sets than what seems to be the prevalent characterization of the subject. This same analysis directs my approach to questions about certain undecidable statements in set theory. My aim is to suggest that a proper understanding of set theory should provoke a realist attitude toward mathematical truth. One consequence of having such an attitude is that the present formal structure of set theory necessarily fails to capture all that there is to know about sets. With that in mind, I propose a change in the foundations of the theory which I hope is a beginning of a richer set theory. This volume was written under the generous support of the Rice Undergraduate Scholars' Program in 1999 and 2000. I thank Professors James Kinsey, Scott Derrick, and Don Johnson for their enthusiasm and the opportunities offered in the program. Thanks are also due to Byeong Uk-Yi and Alasdair Urquhart, whose insights and encouragement greatly affected the direction of my research. Professor Richard Grandy of the Rice University Philosophy Department was my advisor during the course of this research. I am greatly indebted to his contributions in this project.