The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal,9783110157086

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

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ISBN13: 9783110157086
ISBN10: 311015708X
Format: Hardcover
Pub. Date: 1999-07-01
Publisher(s): De Gruyter
List Price: $228.00

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Table of Contents

1 Introduction
1(17)
1.1 The nonstationary ideal on (Omega)(1)
2(4)
1.2 The partial order P(max)
6(4)
1.3 P(max) variations
10(3)
1.4 Extensions of inner models beyond L(R)
13(1)
1.5 Concluding remarks
14(4)
2 Preliminaries
18(30)
2.1 Weakly homogeneous trees and scales
18(10)
2.2 Generic absoluteness
28(3)
2.3 The stationary tower
31(2)
2.4 Forcing Axioms
33(4)
2.5 Reflection Principles
37(3)
2.6 Generic ideals
40(8)
3 The nonstationary ideal
48(67)
3.1 The nonstationary ideal and (Delta)^(1)(2)
48(58)
3.2 The nonstationary ideal and CH
106(9)
4 The P(max)-extension
115(69)
4.1 Iterable structures
115(20)
4.2 The partial order P(max)
135(49)
5 Applications
184(107)
5.1 The sentence (Phi)AC
184(3)
5.2 Martin's Maximum, (Phi)AC and (XXX)(Omega)((Omega)(2))
187(10)
5.3 The sentence (Psi)AC
197(7)
5.4 The stationary tower and P(max)
204(22)
5.5 P^(*)max
226(11)
5.6 P^(0)max
237(6)
5.7 The Axiom (XXX)
243(36)
5.8 Homogeneity properties of (Rho)((Omega)(1))/(XXX)(NS)
279(12)
6 P(max) variations
291(141)
6.1 ^(2)P(max)
292(18)
6.2 Variations for obtaining (Omega)(1) -dense ideals
310(117)
6.2.1 Qmax
310(28)
6.2.2 Q^(*)max
338(36)
6.2.3 ^(2)Qmax
374(7)
6.2.4 Weak Kurepa trees and Qmax
381(7)
6.2.5 ^(KT)Qmax
388(21)
6.2.6 Null sets and the nonstationary ideal
409(18)
6.3 Nonregular ultrafilters on (Omega)(1)
427(5)
7 Conditional variations
432(66)
7.1 Suslin trees
432(15)
7.2 The Borel Conjecture
447(51)
8 (XXX) principles for (Omega)(1)
498(119)
8.1 Condensation Principles
501(6)
8.2 P^(XXX)(NS)(max)
507(78)
8.3 The principles, (XXX)^(+)(NS) and (XXX)^(++)(NS)
585(32)
9 Extensions of L((Gamma), R))
617(152)
9.1 AD^(+)
618(8)
9.2 The P(max)-extension of L((Gamma), R))
626(20)
9.2.1 The basic analysis
627(4)
9.2.2 Martin's Maximum^(++)(c)
631(11)
9.2.3 (XXX)(Omega)((Omega(2))
642(4)
9.3 The Qmax -extension of L((Gamma), R)
646(3)
9.4 Chang's Conjecture
649(15)
9.5 Weak and Strong Reflection Principles
664(16)
9.6 Strong Chang's Conjecture
680(17)
9.7 Ideals on (Omega)(2)
697(72)
10 Further results
769(152)
10.1 Forcing notions and large cardinals
769(7)
10.2 Coding into L((Rho)(Omega)(1)))
776(86)
10.2.1 Coding by sets, S^(XXX)
778(5)
10.2.2 Q^(X)max
783(32)
10.2.3 P^(XXX)max
815(31)
10.2.4 P^((XXX, B)(max)
846(16)
10.3 Bounded forms of Martin's Maximum
862(24)
10.4 (Omega)-logic
886(8)
10.5 (Omega)-logic and the Continuum Hypothesis
894(14)
10.6 The Axiom (*)^(+)
908(8)
10.7 The Effective Singular Cardinals Hypothesis
916(5)
11 Questions
921(6)
Bibliography 927(4)
Index 931

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