The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

Cover
Title page
Contents
1 Introduction
1.1 The nonstationary ideal on ω₁
1.2 The partial order P_{max}
1.3 P_{max} variations
1.4 Extensions of inner models beyond L(R)
1.5 Conc1uding remarks - the view from Berlin in 1999
1.6 The view from Heidelberg in 2010
2 Preliminaries
2.1 Weakly homogeneous trees and scales
2.2 Generic absoluteness
2.3 The stationary tower
2.4 Forcing Axioms
2.5 Refiection Principles
2.6 Generic ideals
3 The nonstationary ideal
3.1 The nonstationary ideal and δ¹₂
3.2 The nonstationary ideal and CH
4 The P_{max}-extension
4.1 Iterable structures
4.2 The partial order P_{max}
5 Applications
5.1 The sentence Φ_{AC}
5.2 Martin's Maximum and Φ_{AC}
5.3 The sentence Ψ_{AC}
5.4 The stationary tower and P_{max}
5.5 P*_{max}
5.6 P⁰_{max}232
5.7 The Axiom (^*_*)
5.8 Homogeneity properties of P(ω₁)/I_{NS}
6 P_{max} variations
6.1 ²P_{max}
6.2 Variations for obtaining ω₁-dense ideals
6.2.1 Q_{max}
6.2.2 Q*_{max}
6.2.3 ²Q_{max}
6.2.4 Weak Kurepa trees and Q_{max}
6.2.5 ^{KT}Q_{max}
6.2.6 Null sets and the nonstationary ideal
6.3 Nonregular ultrafilters on ω₁
7 Conditional variations
7.1 Suslin trees
7.2 The Borel Conjecture
8 ? principles for ω₁
8.1 Condensation Principles
8.2 P*^{NS}_{max}
8.3 The principles ?^+_[NS} and ?^{++}_{NS}
9 Extensions of L(Γ,R)
9.1 AD+
9.2 The P_{max}-extension of L(Γ,R)
9.2.1 The basic analysis
9.2.2 Martin's Maximum ++(c)
9.3 The Q_{max}-extension of L(Γ,R)
9.4 Chang's Conjecture
9.5 Weak and Strong Refiection Principles
9.6 Strong Chang's Conjecture
9.7 Ideals on ω₂
10 Further results
10.1 Forcing notions and large cardinals
10.2 Coding into L(P(ω₁))
10.2.1 Coding by sets, ~S
10.2.2 Q^{(X)}_{max}
10.2.3 P^{(∅)}_{max}. . . . . . . . (0,B)
10.2.4 P^{(∅,B)}_{max}
10.3 Bounded forms of Martin's Maximum
10.4 Ω-logic
10.5 Ω-logic and the Continuum Hypothesis
10.6 The Axiom (*)^+
10.7 The Effective Singular Cardinals Hypothesis
11 Questions
Bibliography
Index