Handbook of Set Theory

This handbook is the definitive compendium of the methods, results, and current initiatives in modern set theory in all its research directions. Set theory has entered its prime as an advanced and autonomous field of mathematics with foundational significance, and the expanse and variety of this handbook attests to the richness and sophistication of the subject. The chapters are written by acknowledged experts, major research figures in their areas, and they each bring to bear their experience and insights in carefully wrought, self-contained expositions. There is historical depth, elegant development, probing to the frontiers, and prospects for the future. This handbook is essential reading for the aspiring researcher, a pivotal focus for the veteran set theorist, and a massive reference for all those who want to gain a larger sense of the tremendous advances that have been made in the subject, one which first appeared as a foundation of mathematics but in the last several decades has expanded into a broad and far-reaching field with its own self-fueling initiatives.

Résumé
Numbers imitate space, which is of such a di?erent nature Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.

Contenu

Stationary Sets.- Partition Relations.- Coherent Sequences.- Borel Equivalence Relations.- Proper Forcing.- Combinatorial Cardinal Characteristics of the Continuum.- Invariants of Measure and Category.- Constructibility and Class Forcing.- Fine Structure.- ? Fine Structure.- Elementary Embeddings and Algebra.- Iterated Forcing and Elementary Embeddings.- Ideals and Generic Elementary Embeddings.- Cardinal Arithmetic.- Successors of Singular Cardinals.- Prikry-Type Forcings.- Beginning Inner Model Theory.- The Covering Lemma.- An Outline of Inner Model Theory.- A Core Model Toolbox and Guide.- Structural Consequences of AD.- Determinacy in (?).- Large Cardinals from Determinacy.- Forcing over Models of Determinacy.