The Higher Infinite

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. The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. The first of a projected multi-volume series, this book provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contempory research. A "genetic" approach is taken, presenting the subject in the context of its historical development. With hindsight the consequential avenues are pursued and the most elegant or accessible expositions given. With open questions and speculations provided throughout the reader should not only come to appreciate the scope and coherence of the overall enterpreise but also become prepared to pursue research in several specific areas by studying the relevant sections. 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, in particular their Lebesgue measurability. Although formulated at various stages in the development of set theory and with different incentives, the hypotheses were found to form a linear hierarchy reaching up to an inconsistent extension of motivating concepts. All known set-theoretic propositions have been gauged in this hierarchy in terms of consistency strength, and the emerging str- ture of implications provides a remarkably rich, detailed and coherent picture of the strongest propositions of mathematics as embedded in set theory. The ?rst of a projected multi-volume series, this text provides a comp- hensive account of the theory of large cardinals from its beginnings through the developments of the early 1970’s and several of the direct outgrowths leading to the frontiers of current research.