br Set theory, initially built on the Cantorian extension of number into the infinite and the Zermelian axiomatization affirming a foundation for mathematics, is today a rich and sophisticated field of mathematics which, having been invested with model-building techniques of Goedel and Cohen, incorporates into its ongoing investigation of infinite sets a full range of strong hypotheses involving large cardinalities. br This volume serves as a companion to set theory, in that it gathers together essays that illuminate the historical development, philosophical resonances, and current mathematical activity. Part I, History, has essays on the development and ..

The higher in?nite refers to the lofty reaches of the in?nite cardinalities of set t- ory as charted out by large cardinal hypotheses. These hypotheses posit cardinals that prescribe their own transcendence over smaller cardinals and provide a sup- structure for the analysis of strong propositions. As such they are the rightful heirs to the two main legacies of Georg Cantor, founder of set theory: the extension of number into the in?nite and the investigation of de?nable sets of reals. The investigation of large cardinal hypotheses is indeed a mainstream of modern set theory, and they have been found to play a crucial role in the study of de?nable sets of reals, ..