Title: Geometric Set Theory
Description: This book introduces a new research direction in set theory: the study of models of set theory with respect to their extensional overlap or disagreement. In Part I, the method is applied to isolate new distinctions between Borel equivalence relations. Part II contains applications to independence results in Zermelo-Fraenkel set theory without Axiom of Choice. The method makes it possible to classify in great detail various paradoxical objects obtained using the Axiom of Choice; the classifying criterion is a ZF-provable implication between the existence of such objects. The book considers a broad spectrum of objects from analysis, algebra, and combinatorics: ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and many more.
Subjects: Mathematics, Descriptive set theory, Equivalence relations (Set theory), Borel sets, Independence (Mathematics), Axiomatic set theory, Forcing (Model theory), Mathematical logic and foundations -- Set theory -- Descriptive set theory, Mathematical logic and foundations -- Set theory -- Axiom of choice and related propositions, Mathematical logic and foundations -- Set theory -- Consistency and independence results, Mathematical logic and foundations -- Set theory -- Other aspects of forcing and Boolean-valued models, Combinatorics -- Graph theory -- Coloring of graphs and hypergraphs, Combinatorics -- Designs and configurations -- Matroids, geometric lattices, Number theory -- Diophantine approximation, transcendental number theory-- Irrationality; linear independence over a field, Number theory -- Diophantine approximation, transcendental number theory -- Transcendence (general theory), Dynamical systems and ergodic theory -- Ergodic theory -- Orbit equivalence, cocycles, ergodic equivalence relations