In this volume the author develops and applies methods for proving, from large cardinals, the determinacy of definable games of countable length on natural numbers. The determinacy is ultimately derived from iteration strategies, connecting games on natural numbers with the specific iteration games that come up in the study of large cardinals. The games considered in this text range in strength, from games of fixed countable length, through games where the length is clocked by natural numbers, to games in which a run is complete when its length is uncountable in an inner model (or a pointclass) relative to the run. More can be done using the methods developed here, reaching determinacy for games of certain length. The book is largely self-contained. Only graduate level knowledge of modern techniques in large cardinals and basic forcing is assumed. Several exercises allow the reader to build on the results in the text, for example connecting them with universally Baire and homogeneously Suslin sets. - Important contribution to one of the main features of current set theory, as initiated and developed by Jensen, Woodin, Steel and others.
Author(s): Itay Neeman
Series: De Gruyter Series in Logic and Its Applications
Publisher: Walter De Gruyter Inc
Year: 2004
Language: English
Pages: 330
Preface......Page 8
Contents......Page 10
Introduction......Page 14
1. Basic components......Page 28
A. The auxiliary games map......Page 29
B. Generic runs......Page 36
C. Pivots......Page 40
D. Mirror images......Page 50
E. Sample application......Page 51
F. Mixed pivots......Page 56
A. General games and iteration games......Page 64
B. Limits......Page 68
C. Successors......Page 81
D. Limits again......Page 86
E. Universally Baire sets......Page 94
A. Codes......Page 100
B. First determinacy result, part I......Page 101
C. First determinacy result, part II......Page 122
D. A slight improvement......Page 132
E. Variation......Page 137
A. Codes......Page 148
B. Woodin's extender algebra......Page 154
C. Names, part I......Page 168
D. Names, part II......Page 175
E. Mirror images......Page 180
A. Saturation......Page 188
B. Successors, basic step......Page 194
C. Relative limits......Page 196
D. Woodin limits of Woodin cardinals......Page 207
E. Relative successors......Page 216
F. Compositions......Page 218
G. Conclusion......Page 219
A. The game......Page 222
B. Successors, basic step......Page 233
C. Relative limits......Page 234
D. Woodin limits of Woodin cardinals......Page 248
E. Relative successors and compositions......Page 257
F. Skips......Page 261
G. Conclusion......Page 278
A. Shifted payoff......Page 281
B. Layout......Page 283
C. Basic step......Page 290
D. Construction......Page 297
E. The main theorem......Page 302
F. Determinacy......Page 305
Extenders......Page 314
Interpretation of K in generic extensions......Page 315
Iteration trees......Page 316
Iterability......Page 318
Bibliography......Page 322
Index......Page 326
