Set Theory: Exploring Independence and Truth

Description

This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory. The following topics are covered: • Forcing and constructability • The Solovay-Shelah Theorem i.e. the equiconsistency of ‘every set of reals is Lebesgue measurable’ with one inaccessible cardinal • Fine structure theory and a modern approach to sharps • Jensen’s Covering Lemma • The equivalence of analytic determinacy with sharps • The theory of extenders and iteration trees • A proof of projective determinacy from Woodin cardinals. Set Theory requires only a basic knowledge of mathematical logic and will be suitable for advanced students and researchers. Front Matter....Pages i-x Naive Set Theory....Pages 1-8 Axiomatic Set Theory....Pages 9-21 Ordinals....Pages 23-31 Cardinals....Pages 33-65 Constructibility....Pages 67-91 Forcing....Pages 93-126 Descriptive Set Theory....Pages 127-146 Solovay’s Model....Pages 147-164 The Raisonnier Filter....Pages 165-182 Measurable Cardinals....Pages 183-233 $$0^{\#}$$ 0 # and Jensen’s Covering Lemma....Pages 235-278 Analytic and Full Determinacy....Pages 279-302 Projective Determinacy....Pages 303-323 Back Matter....Pages 325-332