This handbook is the definitive compendium of the methods, results, and current initiatives in modern set theory in all its research directions. Set theory has entered its prime as an advanced and autonomous field of mathematics with foundational significance, and the expanse and variety of this handbook attests to the richness and sophistication of the subject. The chapters are written by acknowledged experts, major research figures in their areas, and they each bring to bear their experience and insights in carefully wrought, self-contained expositions. There is historical depth, elegant development, probing to the frontiers, and prospects for the future. This handbook is essential reading for the aspiring researcher, a pivotal focus for the veteran set theorist, and a massive reference for all those who want to gain a larger sense of the tremendous advances that have been made in the subject, one which first appeared as a foundation of mathematics but in the last several decades has expanded into a broad and far-reaching field with its own self-fueling initiatives.
Author(s): Matthew Foreman Akihiro Kanamori
Publisher: Springer
Year: 2010
Language: English
Pages: 2200
Introduction......Page 15
Cantor......Page 16
Zermelo......Page 18
First Developments......Page 20
Replacement and Foundation......Page 24
Gödel......Page 27
Infinite Combinatorics......Page 31
Definability......Page 33
Model-Theoretic Techniques......Page 36
Cohen......Page 41
Method of Forcing......Page 43
0#, L[U], and L[U]......Page 47
Constructibility......Page 50
Large Large Cardinals......Page 55
Determinacy......Page 58
Silver's Theorem and Covering......Page 62
Forcing Consistency Results......Page 66
Into the 1980s......Page 70
Consistency of Determinacy......Page 74
Later Developments......Page 77
Summaries of the Handbook Chapters......Page 82
Stationary Sets......Page 107
Closed Unbounded Sets......Page 108
Splitting Stationary Sets......Page 110
Generic Ultrapowers......Page 111
Stationary Sets in Generic Extensions......Page 113
Reflecting Stationary Sets......Page 114
A Hierarchy of Stationary Sets......Page 117
Canonical Stationary Sets......Page 118
Full Reflection......Page 119
kappa+-saturation......Page 120
Precipitousness......Page 123
Closed Unbounded Sets in PkappaA......Page 124
Splitting Stationary Sets......Page 127
Saturation......Page 128
Proper Forcing......Page 129
Projective and Cohen Boolean Algebras......Page 130
Reflection Principles......Page 132
Stationary Tower Forcing......Page 135
Bibliography......Page 136
Partition Relations......Page 143
Introduction......Page 144
Basic Definitions......Page 146
Ramsey's Theorem......Page 148
Ramification Arguments......Page 149
Partition Relations and Submodels......Page 151
Overview......Page 154
More Elementary Submodels......Page 157
The Balanced Generalization......Page 158
The Unbalanced Generalization......Page 161
The Baumgartner-Hajnal Theorem......Page 166
The Milner-Rado Paradox and Omega(kappa)......Page 174
Shelah's Theorem for Infinitely Many Colors......Page 176
Singular Cardinal Resources......Page 179
Successors of Weakly Compact Cardinals......Page 181
Successors of Singular Cardinals......Page 185
Some History......Page 190
Small Counterexamples......Page 191
A Positive Countable Partition Relation......Page 201
Representation......Page 202
Node Labeled Trees......Page 206
Game......Page 208
Uniformization......Page 210
Triangles......Page 213
Free Sets......Page 218
Bibliography......Page 223
Coherent Sequences......Page 228
The Space of Countable Ordinals......Page 230
Subadditive Functions......Page 237
Steps and Coherence......Page 244
The Trace and the Square-Bracket Operation......Page 249
A Square-Bracket Operation on a Tree......Page 256
Special Trees and Mahlo Cardinals......Page 257
The Weight Function on Successor Cardinals......Page 261
The Number of Steps......Page 263
Square Sequences......Page 265
The Full Lower Trace of a Square Sequence......Page 270
Special Square Sequences......Page 273
Successors of Regular Cardinals......Page 276
Successors of Singular Cardinals......Page 280
The Oscillation Mapping......Page 286
The Square-Bracket Operation......Page 288
Unbounded Functions on Successors of Regular Cardinals......Page 294
Higher Dimensions......Page 299
Bibliography......Page 305
Borel Equivalence Relations......Page 310
Definitions......Page 311
Structure......Page 319
Anti-Structure......Page 323
Beyond Good and Evil......Page 324
Countable Borel Equivalence Relations......Page 326
The Global Structure......Page 327
Treeable Equivalence Relations......Page 334
Hyperfiniteness......Page 335
Effective Cardinality......Page 336
Classification Problems......Page 337
Universal for Polish Group Actions......Page 338
Universal for Sinfty......Page 339
E0......Page 340
Bibliography......Page 341
Proper Forcing......Page 346
Introduction......Page 347
Countable Support Iterations......Page 352
Properness and Its Iteration......Page 353
Preservation of Properness......Page 357
The 2-Chain Condition......Page 360
Equivalent Formulations......Page 363
Preservation of omegaomega-Boundedness......Page 365
Application: Non-Isomorphism of Ultrapowers......Page 369
Preservation of Unboundedness......Page 374
The Almost Bounding Property......Page 376
Application to Cardinal Invariants......Page 377
alpha-Properness......Page 382
Equivalent Definition......Page 383
Preservation of alpha-Properness......Page 384
A Coloring Problem......Page 387
Dee-Completeness......Page 392
Two-Step Iteration......Page 396
Proof of Theorem 5.17......Page 398
Simple Completeness Systems......Page 401
The Properness Isomorphism Condition......Page 403
Bibliography......Page 406
Combinatorial Cardinal Characteristics of the Continuum......Page 408
Introduction......Page 409
Growth of Functions......Page 411
Splitting and Homogeneity......Page 415
Galois-Tukey Connections and Duality......Page 421
Category and Measure......Page 430
Sparse Sets of Integers......Page 438
Forcing Axioms......Page 447
Almost Disjoint and Independent Families......Page 455
Filters and Ultrafilters......Page 459
Evasion and Prediction......Page 471
Forcing......Page 481
Countable Support Proper Iteration......Page 483
Cohen Reals......Page 485
Random Reals......Page 486
Sacks Reals......Page 488
Hechler Reals......Page 490
Mathias Reals......Page 491
Miller Reals......Page 492
Other Forcing Iterations......Page 493
Adding One Real......Page 495
Bibliography......Page 496
Invariants of Measure and Category......Page 503
Introduction......Page 504
Tukey Connections......Page 505
Inequalities Provable in ZFC......Page 507
Combinatorial Characterizations......Page 519
Cofinality of cov(J) and COV(J)......Page 530
Consistency Results and Counterexamples......Page 547
Further Reading......Page 560
Bibliography......Page 564
Constructibility and Class Forcing......Page 568
Three Problems of Solovay......Page 570
Tameness......Page 572
Examples......Page 576
Relevance......Page 580
The Coding Theorem......Page 588
The Solovay Problems......Page 599
Generic Saturation......Page 602
Further Results......Page 605
Some Open Problems......Page 613
Bibliography......Page 614
Fine Structure......Page 616
Acceptable J-structures......Page 617
The First Projectum......Page 629
Downward Extension of Embeddings......Page 632
Upward Extension of Embeddings......Page 636
Iterated Projecta......Page 642
Standard Parameters......Page 646
Solidity Witnesses......Page 649
Fine Ultrapowers......Page 652
Applications to L......Page 662
Bibliography......Page 666
Sigma Fine Structure......Page 668
Sigma Fine Structure......Page 669
Variant Fine Structures......Page 682
Sigma Ultrapowers......Page 684
Pseudo-Ultrapowers......Page 696
Global......Page 703
Defining Cnu......Page 712
Variants and Generalities on......Page 725
in Fine-Structural Inner Models......Page 731
Morasses......Page 735
Construction of Gap-1 Morasses in L......Page 737
Variants......Page 741
Bibliography......Page 744
Elementary Embeddings and Algebra......Page 748
Kunen's Bound and Axiom (I3)......Page 749
Operations on Elementary Embeddings......Page 751
Iterations of an Elementary Embedding......Page 753
Finite Quotients......Page 756
The Laver-Steel Theorem......Page 760
Counting the Critical Ordinals......Page 762
Iterated Left Division in LD-systems......Page 767
Using Elementary Embeddings......Page 770
Avoiding Elementary Embeddings......Page 771
Finite LD-Systems......Page 773
Using Elementary Embeddings......Page 776
Not Avoiding Elementary Embeddings?......Page 779
Bibliography......Page 784
Iterated Forcing and Elementary Embeddings......Page 786
Introduction......Page 787
Elementary Embeddings......Page 792
Ultrapowers and Extenders......Page 795
Large Cardinal Axioms......Page 798
Forcing......Page 800
Some Forcing Posets......Page 805
Iterated Forcing......Page 809
Building Generic Objects......Page 814
Lifting Elementary Embeddings......Page 817
Generic Embeddings......Page 820
Iteration with Easton Support......Page 822
Master Conditions......Page 825
A Technique of Magidor......Page 830
Absorption......Page 831
Transfer and Pullback......Page 839
Small Large Cardinals......Page 840
A Precipitous Ideal on omega1......Page 842
Iterated Club Shooting......Page 844
Details......Page 845
Precipitousness......Page 848
A Lower Bound......Page 851
Precipitousness for NSomega2 | Cof(omega1)......Page 852
Outline of the Proof and Main Technical Issues......Page 853
Namba Forcing, RCS Iteration and the S and I Conditions......Page 855
The Preparation Iteration......Page 857
A Warm-up for the Main Iteration......Page 858
The Main Iteration......Page 863
Precipitousness of the Non-Stationary Ideal......Page 865
Successors of Larger Cardinals......Page 866
More on Iterated Club Shooting......Page 867
More on Collapses......Page 871
Limiting Results......Page 873
Termspace Forcing......Page 876
More on Termspace Forcing and Collapsing......Page 879
Iterations with Prediction......Page 882
Altering Generic Objects......Page 888
Bibliography......Page 890
Ideals and Generic Elementary Embeddings......Page 895
Introduction......Page 897
An Overview of the Chapter......Page 898
Apology and Acknowledgments......Page 900
Some Conventions Used in the Chapter......Page 901
Basic Facts......Page 902
The Generic Ultrapower......Page 904
Disjointing Property and Closure of Ultrapowers......Page 908
Normal Ideals......Page 910
More General Facts......Page 914
Canonical Functions......Page 916
Ideals and Reflection......Page 918
Examples......Page 920
The Closed Unbounded Filter and the Nonstationary Ideal......Page 921
Natural Ideals on P(R)......Page 927
I[lambda] and Related Ideals......Page 928
Club Guessing Ideals......Page 930
Ideals of Sets Without Guessing Sequences......Page 931
Uniformization Ideals......Page 933
Induced Ideals......Page 934
General Induced Ideals......Page 937
Goodness and Self-Genericity......Page 938
A Structural Property of Saturated Ideals......Page 941
Saturation Properties......Page 942
Layered Ideals......Page 947
Projections......Page 951
Where the Ordinals Go......Page 955
A Discussion of Large Sets......Page 958
Iterating Ideals......Page 962
Generic Ultrapowers by Towers......Page 964
Using Reflection......Page 968
Chang's Conjectures, Jónsson Cardinals and Square......Page 970
Ideals and GCH......Page 972
Woodin's Theorem Showing CH Holds......Page 973
Abe's Results on SCH......Page 977
The Value of Theta......Page 978
Stationary Set Reflection......Page 981
Suslin and Kurepa Trees......Page 983
Partition Properties......Page 985
The Normal Moore Space Conjecture and Variants......Page 989
Consequences in Descriptive Set Theory......Page 990
Connections with Non-Regular Ultrafilters......Page 991
Fully Non-Regular Ultrafilters on omega2......Page 994
Graphs and Chromatic Numbers......Page 995
The Nonstationary Ideal on omega1......Page 996
Some Limitations......Page 999
Soft Limitations......Page 1000
The "Kunen Argument"......Page 1002
Saturated Ideals and Cofinalities......Page 1004
Closed Unbounded Subsets of [kappa]omega......Page 1008
Uniform Ideals on Ordinals......Page 1009
Restrictions on the Quotient Algebra......Page 1011
Yet Another Result of Kunen......Page 1017
The Matsubara-Shioya Theorem......Page 1018
The Nonstationary Ideal on [lambda]Trivial Master Conditions......Page 1024
The Basic Idea......Page 1025
Precipitous Ideals on Accessible Cardinals......Page 1027
Strong Master Conditions......Page 1028
Precipitousness is not Preserved Under Projections......Page 1029
Computing Quotient Algebras and Preserving Strong Ideals under Generic Extensions......Page 1030
Preservation of Normality and a Warm-Up Theorem......Page 1031
When Master Conditions Exist......Page 1033
The Duality Theorem and Its Consequences......Page 1034
Understanding the Embeddings: Master Conditions Must Exist......Page 1037
Pseudo-Generic Towers......Page 1039
A kappa-Saturated Ideal on an Inaccessible Cardinal kappa......Page 1040
Basic Kunen Technique: kappa+-Saturated Ideals......Page 1042
(2,2,0)-Saturated Ideals......Page 1045
Chang Ideals with Simple Quotients I......Page 1047
3-Dense Ideals on omega3......Page 1048
Higher Chang's Conjectures and omega Jónsson......Page 1049
The Magidor Variation......Page 1050
More Saturated Ideals......Page 1052
Forbidden Intervals......Page 1059
Dense Ideals on omega1......Page 1060
The Lower End......Page 1062
Chang-Type Ideals with Simple Quotients II......Page 1067
Destroying Precipitous and Saturated Ideals......Page 1068
Forcing over Determinacy Models......Page 1069
Making Induced Ideals Natural......Page 1070
The Null and Meager Ideals......Page 1071
Nonstationary Ideals......Page 1072
Club Guessing Ideals......Page 1076
Making Natural Ideals Have Well-Founded Ultrapowers......Page 1081
Catching Antichains Using Reflection......Page 1083
Reflecting Stationary Sets......Page 1085
Catching Your Tail......Page 1088
The Nonstationary Ideal is Precipitous......Page 1089
Saturation of the Nonstationary Ideal......Page 1090
The Equivalence of "Semiproper" with "Stationary Set Preserving" in the Case of Antichain Sealing Forcing......Page 1092
Martin's Maximum and Related Topics......Page 1095
Shelah's Results on Ulam's Problem......Page 1097
Saturated Ideals and Square......Page 1098
Tower Forcing......Page 1099
Induced Towers......Page 1101
Good Structures......Page 1103
Catching Antichains......Page 1104
Catching Your Tail......Page 1105
Using Antichain Catching......Page 1106
Methods for Proving Antichain Capturing......Page 1109
Woodin's Towers......Page 1111
Burke's Towers......Page 1114
Self-Genericity for Towers......Page 1117
Examples of Stationary Tower Forcing......Page 1119
A Tower that is not Precipitous......Page 1124
Fine Structural Inner Models......Page 1126
Constructing from Stationary Sets and the Nonstationary Ideal......Page 1128
Decisive Ideals......Page 1129
Chang's Conjecture and Huge Cardinals......Page 1131
A Martin's Maximum Result......Page 1133
Ideals as Axioms......Page 1134
Generalized Large Cardinals......Page 1135
Flies in the Ointment......Page 1136
Some Examples of Axioms......Page 1138
Predictions......Page 1140
Gradations of Consequence......Page 1142
A Final Relevant Issue......Page 1143
Open Questions......Page 1144
Bibliography......Page 1150
Cardinal Arithmetic......Page 1158
Introduction......Page 1159
Elementary Definitions......Page 1160
Partial Orderings......Page 1162
Projections......Page 1165
Existence of Exact Upper Bounds......Page 1167
Application: Silver's Theorem......Page 1179
Application: A Covering Theorem......Page 1180
Basic Properties of the pcf Function......Page 1182
The Ideal JGenerators for JThe Cofinality of [µ]kappa......Page 1196
Elementary Substructures......Page 1197
Minimally Obedient Sequences......Page 1200
Application: The Cofinality of ([µ]kappa, )......Page 1205
Elevations and Transitive Generators......Page 1211
Localization......Page 1214
Size Limitation on pcf of Intervals......Page 1217
Revised GCH......Page 1219
TD(f)......Page 1225
Proof of the Revised GCH......Page 1228
Applications of the Revised GCH......Page 1229
Bibliography......Page 1235
Successors of Singular Cardinals......Page 1237
Reflection of Stationary Sets......Page 1238
Is omega+1 a Jónsson cardinal?......Page 1239
Elementary Submodels......Page 1241
pcf Theory......Page 1243
Large Cardinals......Page 1247
Remarks......Page 1248
On Stationary Reflection......Page 1249
Squares and Supercompact Cardinals......Page 1250
Reflection and Indecomposable Ultrafilters......Page 1253
Reflection at omega+1-Introduction......Page 1256
kappa+-closed Forcing and Stationary Subsets of Slambdakappa......Page 1257
Reflection at omega+1-Conclusion......Page 1261
On I[lambda]......Page 1263
The Ideal I[lambda]......Page 1264
Construction......Page 1269
The Approachability Property......Page 1270
The Extent of I[lambda]......Page 1274
Weak Approachability and APomega......Page 1277
The Structure of I[lambda]......Page 1282
An Application-the Existence of Scales......Page 1288
Weakenings of -Part I......Page 1295
Weakenings of -Part II......Page 1298
On......Page 1304
Very Weak Square......Page 1309
NPT and Good Scales......Page 1314
Varieties of Nice Scales......Page 1325
Some Consequences of pp(µ)>µ+......Page 1332
Trees at Successors of Singular Cardinals......Page 1335
Square-Bracket Partition Relations......Page 1336
Colorings of Finite Subsets......Page 1337
Colorings of Pairs......Page 1341
Colorings and Club Guessing......Page 1346
Concluding Remarks......Page 1352
Bibliography......Page 1353
Prikry-Type Forcings......Page 1359
Basic Prikry Forcing......Page 1360
Tree Prikry Forcing......Page 1364
Adding a Prikry Sequence to a Singular Cardinal......Page 1367
Supercompact and Strongly Compact Prikry Forcings......Page 1369
Adding Many Prikry Sequences to a Singular Cardinal......Page 1373
Extender-Based Prikry Forcing with a Single Extender......Page 1386
Down to omega......Page 1399
Forcing Uncountable Cofinalities......Page 1407
Radin Forcing......Page 1408
Magidor Forcing and Coherent Sequences of Measures......Page 1425
Extender-Based Radin Forcing......Page 1427
Magidor Iteration......Page 1432
Leaning's Forcing......Page 1440
Easton Support Iteration......Page 1441
Limit Levels......Page 1446
An Application to Distributive Forcing Notions......Page 1449
Some Open Problems......Page 1451
Bibliography......Page 1452
Beginning Inner Model Theory......Page 1456
The Constructible Sets......Page 1458
Relative Constructibility......Page 1460
Measurable Cardinals......Page 1461
0#, and Sharps in General......Page 1465
Other Sharps......Page 1468
From Sharps to the Core Model......Page 1470
Beyond One Measurable Cardinal......Page 1471
The Comparison Process......Page 1473
Indiscernibles from Iterated Ultrapowers......Page 1478
Extender Models......Page 1480
The Modern Presentation of L[E]......Page 1491
Remarks on Larger Cardinals......Page 1493
Woodin cardinals......Page 1494
Supercompact Cardinals......Page 1495
What is the Core Model?......Page 1496
Bibliography......Page 1500
The Covering Lemma......Page 1503
The Statement......Page 1504
The Weak Covering Lemma......Page 1507
The Strong Covering Lemma......Page 1508
The Covering Lemma without Second-Order Closure......Page 1509
Basic Applications......Page 1510
The Weak Covering Lemma......Page 1511
The Full Covering Lemma......Page 1512
The Proof......Page 1514
Fine Structure and Other Tools......Page 1515
Embeddings of Mice......Page 1521
Proof of the Covering Lemma for L......Page 1523
Suitable Sets......Page 1525
Measurable Cardinals......Page 1529
Comparisons of Mice......Page 1539
The Dodd-Jensen Core Model......Page 1544
Part 1 of the Proof......Page 1548
Part 2 of the Proof: Analyzing the Indiscernibles......Page 1553
Unsuitable Covering Sets......Page 1557
The Covering Lemma and Sequences of Measures......Page 1560
Extenders......Page 1563
The Core Model for Sequences of Measures......Page 1564
The Covering Lemma up to o(kappa)=kappa++......Page 1570
Introduction to the Proof......Page 1573
Part 1 of the Proof......Page 1574
Part 2 of the Proof: Analyzing the Indiscernibles......Page 1576
Digression for Non-Countably Closed Sets X......Page 1578
Continuation of the Main Proof......Page 1580
The Singular Cardinal Hypothesis......Page 1585
Up to a Strong Cardinal......Page 1590
Up to a Woodin Cardinal......Page 1594
Bibliography......Page 1596
An Outline of Inner Model Theory......Page 1601
Premice......Page 1602
Extenders......Page 1603
Fine Extender Sequences......Page 1605
The Levy Hierarchy, Cores, and Soundness......Page 1609
Fine Structure and Ultrapowers......Page 1615
Iteration Trees and Comparison......Page 1617
Iteration Trees......Page 1618
The Comparison Process......Page 1623
The Copying Construction......Page 1628
The Dodd-Jensen Lemma......Page 1632
The Weak Dodd-Jensen Property......Page 1634
Solidity and Condensation......Page 1637
Kc-Constructions......Page 1644
The Iterability of Kc......Page 1647
Large Cardinals in Kc......Page 1655
The Reals of Momega......Page 1657
Iteration Strategies in L(R)......Page 1658
Correctness and Genericity Iterations......Page 1663
HODL(R) below Theta......Page 1674
Bibliography......Page 1688
A Core Model Toolbox and Guide......Page 1691
Introduction......Page 1692
Second-Order Definition of K......Page 1694
First-Order Definition of K......Page 1713
Covering Properties......Page 1722
Absoluteness, Complexity and Correctness......Page 1723
Embeddings of K......Page 1724
Combinatorial Principles......Page 1725
On the Technical Hypothesis......Page 1726
Proof of Weak Covering......Page 1727
Determinacy......Page 1743
Tree Representations and Absoluteness......Page 1746
Ideals and Generic Embeddings......Page 1748
Square and Aronszajn Trees......Page 1749
Forcing Axioms......Page 1751
The Failure of UBH......Page 1753
Bibliography......Page 1754
Structural Consequences of AD......Page 1758
Introduction......Page 1759
Prewellordering, Scales, and Periodicity......Page 1765
Projective Ordinals, Sets, and the Coding Lemma......Page 1774
Wadge Degrees and Abstract Pointclasses......Page 1778
The Scale Theory of L(R)......Page 1783
Determinacy and Coding Results......Page 1784
Partition Relations......Page 1787
Suslin Cardinals......Page 1790
Pointclass Arguments......Page 1791
The Next Suslin Cardinal......Page 1798
More on Lambda in the Type IV Case......Page 1806
The Classification of the Suslin Cardinals......Page 1809
Trivial Descriptions: A Theory of omega1......Page 1811
Analysis of Measures on delta11......Page 1815
The Strong Partition Relation on omega1......Page 1818
The Weak Partition Relation on delta13......Page 1822
The Kechris-Martin Theorem Revisited......Page 1834
Martin's Theorem on Normal Measures......Page 1841
Some Canonical Measures......Page 1852
The Higher Descriptions......Page 1854
Some Further Results......Page 1865
Global Results......Page 1866
Generic Codes......Page 1867
Weak Square and Uniform Cofinalities......Page 1871
Some Final Remarks......Page 1876
Bibliography......Page 1878
Determinacy in L(R)......Page 1882
Extenders and Iteration Trees......Page 1886
Iterability......Page 1894
Creating Iteration Trees......Page 1901
Homogeneously Suslin Sets......Page 1908
Projections and Complementations......Page 1914
Universally Baire Sets......Page 1926
Genericity Iterations......Page 1933
Determinacy in L(R)......Page 1945
Bibliography......Page 1953
Large Cardinals from Determinacy......Page 1956
A. Determinacy......Page 1957
B. Large Cardinals......Page 1960
C. Determinacy from Large Cardinals......Page 1962
D. Large Cardinals from Determinacy......Page 1963
E. Overview......Page 1966
Notation......Page 1968
Preliminaries......Page 1970
Boundedness and Basic Coding......Page 1973
Measurability......Page 1977
The Least Stable......Page 1982
Measurability of the Least Stable......Page 1989
Coding Lemma......Page 1992
Uniform Coding Lemma......Page 1996
Applications......Page 2001
A Woodin Cardinal in HODL(R)......Page 2004
Reflection......Page 2005
Strong Normality......Page 2016
A Woodin Cardinal......Page 2032
Woodin Cardinals in General Settings......Page 2036
First Abstraction......Page 2038
Strategic Determinacy......Page 2040
Generation Theorem......Page 2048
Special Cases......Page 2072
Definable Determinacy......Page 2079
Lightface Definable Determinacy......Page 2080
Boldface Definable Determinacy......Page 2099
Second-Order Arithmetic......Page 2106
First Localization......Page 2107
Second Localization......Page 2113
Large Cardinals and Determinacy......Page 2114
HOD-Analysis......Page 2118
Bibliography......Page 2124
Forcing over Models of Determinacy......Page 2125
Iterations......Page 2127
Pmax......Page 2134
Sequences of Models and Countable Closure......Page 2138
Generalized Iterability......Page 2141
The Basic Analysis......Page 2149
psiAC and the Axiom of Choice......Page 2155
Maximality and Minimality......Page 2156
Larger Models......Page 2163
Omega-Logic......Page 2166
Variations for NSomega1......Page 2171
Conditional Variations for Sigma2 sentences......Page 2175
Bibliography......Page 2178
