This book explores a new research direction in set theory, studying models of set theory based on their extensional overlap or disagreement. It applies the method to isolate new distinctions between Borel equivalence relations and applies it to independence results in Zermelo-Fraenkel set theory without the Axiom of Choice. The method allows for the classification of paradoxical objects obtained using the Axiom of Choice, and the book covers a wide range of objects from analysis, algebra, and combinatorics. Format: Paperback / softback Length: 340 pages Publication date: 30 September 2020 Publisher: American Mathematical Society This groundbreaking book delves into a novel research domain within set theory, exploring the study of models of set theory concerning their extensional overlap or disagreement. In Part I, the method is applied to identify fresh distinctions between Borel equivalence relations, shedding light on the intricate relationships between sets. Part II showcases its applications to independence results in Zermelo-Fraenkel set theory, even without the Axiom of Choice. By employing this method, the book enables the precise classification of a wide array of paradoxical objects derived from the Axiom of Choice. The classifying criterion revolves around a ZF-provable implication, which establishes the existence of these peculiar entities. This comprehensive exploration encompasses a diverse range of subjects from analysis, algebra, and combinatorics, including ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and numerous other fascinating topics. The subject matter is vast and brimming with potential for further investigation, making this field an exciting and ever-evolving realm of study. Weight: 614g Dimension: 180 x 256 x 20 (mm) ISBN-13: 9781470454623
