Set Theory

Description

Set Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as a desktop reference. Front Matter....Pages 1-1 Axioms of Set Theory....Pages 3-15 Ordinal Numbers....Pages 17-26 Cardinal Numbers....Pages 27-35 Real Numbers....Pages 37-45 The Axiom of Choice and Cardinal Arithmetic....Pages 47-61 The Axiom of Regularity....Pages 63-71 Filters, Ultrafilters and Boolean Algebras....Pages 73-89 Stationary Sets....Pages 91-105 Combinatorial Set Theory....Pages 107-123 Measurable Cardinals....Pages 125-138 Borel and Analytic Sets....Pages 139-153 Models of Set Theory....Pages 155-172 Front Matter....Pages 173-173 Constructible Sets....Pages 175-200 Forcing....Pages 201-224 Applications of Forcing....Pages 225-265 Iterated Forcing and Martin’s Axiom....Pages 267-283 Large Cardinals....Pages 285-309 Large Cardinals and L ....Pages 311-337 Iterated Ultrapowers and L [ U ]....Pages 339-364 Very Large Cardinals....Pages 365-387 Front Matter....Pages 173-173 Large Cardinals and Forcing....Pages 389-407 Saturated Ideals....Pages 409-439 The Nonstationary Ideal....Pages 441-456 The Singular Cardinal Problem....Pages 457-477 Descriptive Set Theory....Pages 479-509 The Real Line....Pages 511-541 Front Matter....Pages 543-543 Combinatorial Principles in L ....Pages 545-556 More Applications of Forcing....Pages 557-571 More Combinatorial Set Theory....Pages 573-583 Complete Boolean Algebras....Pages 585-599 Proper Forcing....Pages 601-614 More Descriptive Set Theory....Pages 615-626 Determinacy....Pages 627-645 Supercompact Cardinals and the Real Line....Pages 647-658 Inner Models for Large Cardinals....Pages 659-668 Forcing and Large Cardinals....Pages 669-680 Martin’s Maximum....Pages 681-694 More on Stationary Sets....Pages 695-705