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.
Ralf Schindler teaches at Universitat Munster and is an expert in the field of set theory. Ralf Schindler works mostly in the area of descriptive inner model theory. His results are on the construction of inner models and core models, on coding over core models and on applications of inner model theory to descriptive set theory and combinatorics. He isolated the concept of a "remarkable" cardinal.
Naive set theory.- Axiomatic set theory.- Ordinals.- Cardinals.- Constructability.- Forcing.- Descriptive set theory.- Solovay's model.- The Raisonnier filter.- Measurable cardinals.- 0# and Jensen's Covering Lemma.- Analytic and full determinacy.- Projective determinacy.