Extensions of the Axiom of Determinacy

Cover Title page Contents Introduction 1. Notation 2. Prerequisites 3. Forms of Choice 4. Partial orders Part 1. Preliminaries Chapter 1. Determinacy 1. The Axiom of Determinacy 2. Turing Determinacy Chapter 2. The Wadge Hierarchy 1. Lipschitz determinacy 2. Lipschitz degrees and Wadge degrees 3. Pointclasses 4. Universal sets 5. The height of the Wadge hierarchy Chapter 3. Coding Lemmas Chapter 4. Properties of Pointclasses 1. Separation and reduction 2. The prewellordering property 3. Prewellorderings and wellfounded relations 4. Closure under wellordered unions Chapter 5. Strong Partition Cardinals Chapter 6. Suslin Sets and Uniformization 1. Suslin sets 2. Uniformization 3. The Solovay sequence Part 2. AD+ Chapter 7. Ordinal Determinacy Chapter 8. Infinity-Borel Sets 1. ∞-Borel codes 2. Local ∞-Borel codes Chapter 9. Cone Measure Ultraproducts 1. S-cones 2. Measurable cardinals from cone measures 3. The degree order on sets of ordinals 4. Pointed trees 5. Coding ultrapowers 6. Forcing with positive sets Chapter 10. Vopěnka Algebras 1. The Vopěnka algebra 2. Codes for projections, and Uniformization 3. The Vopěnka algebra for ∞-Borel sets Chapter 11. Suslin Sets and Strong Codes 1. Generic codes 2. Producing strong generic codes 3. ∞-Borel representations from Uniformization 4. Closure of the Suslin cardinals Chapter 12. Scales from Uniformization 1. Ordinal determinacy in the codes 2. Boundedness for ∞-Borel relations 3. Becker’s argument Chapter 13. Real Determinacy from Scales 1. Weakly homogeneous trees 2. Normal measures on P_{א1}(λ) 3. AD implies AD_R if all sets of reals are Suslin Questions Bibliography Index Back Cover Cover 1 Title page 4 Contents 6 Introduction 8 1. Notation 11 2. Prerequisites 13 3. Forms of Choice 13 4. Partial orders 13 Part 1. Preliminaries 16 Chapter 1. Determinacy 18 1. The Axiom of Determinacy 18 2. Turing Determinacy 20 Chapter 2. The Wadge Hierarchy 24 1. Lipschitz determinacy 24 2. Lipschitz degrees and Wadge degrees 27 3. Pointclasses 30 4. Universal sets 32 5. The height of the Wadge hierarchy 37 Chapter 3. Coding Lemmas 40 Chapter 4. Properties of Pointclasses 46 1. Separation and reduction 46 2. The prewellordering property 50 3. Prewellorderings and wellfounded relations 55 4. Closure under wellordered unions 57 Chapter 5. Strong Partition Cardinals 60 Chapter 6. Suslin Sets and Uniformization 68 1. Suslin sets 68 2. Uniformization 74 3. The Solovay sequence 75 Part 2. AD+ 78 Chapter 7. Ordinal Determinacy 80 Chapter 8. Infinity-Borel Sets 86 1. ∞-Borel codes 86 2. Local ∞-Borel codes 91 Chapter 9. Cone Measure Ultraproducts 98 1. S-cones 100 2. Measurable cardinals from cone measures 102 3. The degree order on sets of ordinals 103 4. Pointed trees 107 5. Coding ultrapowers 108 6. Forcing with positive sets 110 Chapter 10. Vopěnka Algebras 114 1. The Vopěnka algebra 114 2. Codes for projections, and Uniformization 119 3. The Vopěnka algebra for ∞-Borel sets 123 Chapter 11. Suslin Sets and Strong Codes 130 1. Generic codes 130 2. Producing strong generic codes 134 3. ∞-Borel representations from Uniformization 136 4. Closure of the Suslin cardinals 139 Chapter 12. Scales from Uniformization 142 1. Ordinal determinacy in the codes 142 2. Boundedness for ∞-Borel relations 147 3. Becker’s argument 150 Chapter 13. Real Determinacy from Scales 158 1. Weakly homogeneous trees 158 2. Normal measures on P_{א1}(λ) 164 3. AD implies AD_R if all sets of reals are Suslin 167 Questions 172 Bibliography 174 Index 178 Back Cover 182