Geometric Set Theory

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. The topic is nearly inexhaustible in its variety, and many directions invite further investigation. Contents 6 Preface 10 1. Introduction 12 Part I. Equivalence relations 44 2. The virtual realm 46 3. Turbulence 84 4. Nested sequences of models 104 Part II. Balanced extensions of the Solovay model 114 5. Balanced Suslin forcing 116 6. Simplicial complex forcings 128 7. Ultrafilter forcings 146 8. Other forcings 156 9. Preserving cardinalities 190 10. Uniformization 224 11. Locally countable structures 238 12. The Silver divide 280 13. The arity divide 310 14. Other combinatorics 324 Bibliography 334 Index 340