Geometric Set Theory

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 ?_{?_{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. ?-Bernstein balanced forcing
12.4. Existence of generic filters
Chapter 13. The arity divide
13.1. ?,?-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
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