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. Cover Title page Copyright Contents Preface Chapter 1. Introduction 1.1. Outline of the subject 1.2. Equivalence relation results 1.3. Independence: by topic 1.4. Independence: by model 1.5. Independence: by preservation theorem 1.6. Navigation 1.7. Notation and terminology Part I . Equivalence relations Chapter 2. The virtual realm 2.1. Virtual equivalence classes 2.2. Virtual structures 2.3. Classification: general theorems 2.4. Classification: specific examples 2.5. Cardinal invariants 2.6. Restrictions on partial orders 2.7. Absoluteness 2.8. Dichotomies Chapter 3. Turbulence 3.1. Independent functions 3.2. Examples and operations 3.3. Placid equivalence relations 3.4. Examples and operations 3.5. Absoluteness 3.6. A variation for measure Chapter 4. Nested sequences of models 4.1. Prologue 4.2. Coherent sequences of models 4.3. Choice-coherent sequences of models Part II . Balanced extensions of the Solovay model Chapter 5. Balanced Suslin forcing 5.1. Virtual conditions 5.2. Balanced virtual conditions 5.3. Weakly balanced Suslin forcing Chapter 6. Simplicial complex forcings 6.1. Basic concepts 6.2. Fragmented complexes 6.3. Matroids 6.4. Quotient variations Chapter 7. Ultrafilter forcings 7.1. A Ramsey ultrafilter 7.2. Fubini powers of the Fréchet ideal 7.3. Ramsey sequences of structures 7.4. Semigroup ultrafilters Chapter 8. Other forcings 8.1. Coloring graphs 8.2. Coloring hypergraphs 8.3. Discontinuous homomorphisms 8.4. Automorphisms of power (gw ) modulo finite 8.5. Kurepa families 8.6. Set mappings 8.7. Saturated models on quotient spaces 8.8. Non-DC variations 8.9. Side condition forcings 8.10. Weakly balanced variations Chapter 9. Preserving cardinalities 9.1. The well-ordered divide 9.2. The smooth divide 9.3. The turbulent divide 9.4. The orbit divide 9.5. The E_{K_{gs }} divide 9.6. The pinned divide Chapter 10. Uniformization 10.1. Tethered Suslin forcing 10.2. Uniformization theorems 10.3. Examples Chapter 11. Locally countable structures 11.1. Central objects and notions 11.2. Definable control 11.3. Centered control 11.4. Linked control 11.5. Measured control 11.6. Ramsey control 11.7. Liminf control 11.8. Compactly balanced posets Chapter 12. The Silver divide 12.1. Perfectly balanced forcing 12.2. Bernstein balanced forcing 12.3. n-Bernstein balanced forcing 12.4. Existence of generic filters Chapter 13. The arity divide 13.1. m,n-centered and balanced forcings 13.2. Preservation theorems 13.3. Examples Chapter 14. Other combinatorics 14.1. Maximal almost disjoint families 14.2. Unbounded linear suborders 14.3. Measure and category 14.4. The Ramsey ultrafilter extension Bibliography Index Back Cover