Large Cardinals with Forcing

This chapter describes, following the historical development, the investigation of large cardinal hypotheses using the method of forcing. Large cardinal hypotheses, also regarded as strong axioms of infinity, have stimulated a vast mainstream of modern set theory, and William Mitchell’s chapter in this volume deals with their investigation through inner models, Menachem Kojman’s chapter with their involvement in the study of singular cardinals, and Paul Larson’s chapter with the accentuated direction of determinacy hypotheses. With the subject seen as a larger, integrated whole, we in this chapter incorporate aspects of these others while pursuing the directions involving forcing. The author’s other chapter in this volume, “Set Theory from Cantor to Cohen” (henceforth referred to as CC for convenience), had presented the historical development of set theory through to the creation of the method of forcing. Also, the author’s book, The Higher Infinite [2003], provided the theory of large cardinals, including determinacy, up to a first plateau of combinatorial sophistication. In this chapter we first pick up the historical threads to large cardinals in that other chapter and pursue the developments involving forcing, thereby providing a complementary overlay to the book. There is no attempt at being comprehensive; rather we describe the earlier pivotal results and take up the more persistent themes, those that have mainly driven the subject.