This is the revised and updated second edition of a well-established research monograph on the axiom of determinacy, written by an expert in the field. This axiom is a fundamental statement in set theory, and it is related to winning strategies in game theory.
Author(s): W. Hugh Woodin
Series: De Gruyter Series in Logic and Its Applications
Edition: 2
Publisher: De Gruyter
Year: 2010
Language: English
Pages: 859
Contents......Page 6
1. Introduction......Page 8
1. The nonstationary ideal on ω_1......Page 9
2. The partial order P_max......Page 13
3. P_max variations......Page 17
4. Extensions of inner models beyond L(R)......Page 20
5. Concluding remarks - the view from Berlin in 1999......Page 22
6. The view from Heidelberg in 2010......Page 25
1. Weakly homogeneous trees and scales......Page 28
2. Generic absoluteness......Page 38
3. The stationary tower......Page 41
4. Forcing axioms......Page 43
5. Reflection principles......Page 48
6. Generic ideals......Page 50
1. The nonstationary ideal and δ^1_2......Page 58
2. The nonstationary ideal and CH......Page 115
1. Iterable structures......Page 123
2. The partial order P_max......Page 143
1. The sentence φ_AC......Page 191
2. Martin's Maximum and φ_AC......Page 194
3. The sentence ψ_AC......Page 199
4. The stationary tower and P_max......Page 206
5. P*_max......Page 228
6. P^0_max......Page 239
7. The Axiom (*_*)......Page 245
8. Homogeneity properties of P(ω_1)/J_NS......Page 281
6. P_max variations......Page 294
1. ^2 P_max......Page 295
1. Q_max......Page 313
2. Q*_max......Page 341
3. ^2 Q_max......Page 377
4. Weak Kurepa trees and Q_max......Page 384
5. ^KT Q_max......Page 390
6. Null sets and the nonstationary ideal......Page 410
3. Nonregular ultrafilters on ω_1......Page 428
1. Suslin trees......Page 433
2. The Borel conjecture......Page 448
8. ♣ principles for ω_1......Page 500
1. Condensation principles......Page 503
2. P^{♣_NS}_max......Page 508
3. The principles ♣^+_NS and ♣^++_NS......Page 584
9. Extensions of L(Γ, R)......Page 616
1. AD^+......Page 617
2. The P_max-extension of L(Γ, R)......Page 624
1. The basic analysis......Page 625
2. Martin's Maximum^++(c)......Page 629
3. The Q_max-extension of L(Γ, R)......Page 640
4. Chang's Conjecture......Page 644
5. Weak and strong reflection principles......Page 658
6. Strong Chang's Conjecture......Page 674
7. Ideals on ω_2......Page 690
1. Forcing notions and large cardinals......Page 701
2. Coding into L(P(ω_1))......Page 708
1. Coding by sets, S......Page 710
2. Q^(X)_max......Page 715
3. P^(∅)_max......Page 746
4. P^(∅,B)_max......Page 775
3. Bounded forms of Martin's Maximum......Page 791
4. Ω-logic......Page 814
5. Ω-logic and the Continuum Hypothesis......Page 820
6. The Axiom (*)^+......Page 834
7. The Effective Singular Cardinals Hypothesis......Page 842
11. Questions......Page 847
Bibliography......Page 852
Index......Page 856
