Part I. Games and Scales: 1. Games and scales; Introduction to Part I John R. Steel; 2. Notes on the theory of scales Alexander S. Kechris and Yiannis N. Moschovakis; 3. Propagation of the scale property using games Itay Neeman; 4. Scales on E-sets John R. Steel; 5. Inductive scales on inductive sets Yiannis N. Moschovakis; 6. The extent of scales in L(R) Donald A. Martin and John R. Steel; 7. The largest countable this, that, and the other Donald A. Martin; 8. Scales in L(R) John R. Steel; 9. Scales in K(R) John R. Steel; 10. The real game quantifier propagates scales Donald A. Martin; 11. Long games John R. Steel; 12. The length-w1 open game quantifier propagates scales John R. Steel; Part II. Suslin Cardinals, Partition Properties, Homogeneity: 13. Suslin cardinals, partition properties, homogeneity; Introduction to Part II Steve Jackson; 14. Suslin cardinals, K-suslin sets and the scale property in the hyperprojective hierarchy Alexander S. Kechris; 15. The axiom of determinacy, strong partition properties and nonsingular measures Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis and W. Hugh Woodin; 16. The equivalence of partition properties and determinacy Alexander S. Kechris; 17. Generic codes for uncountable ordinals, partition properties, and elementary embeddings Alexander S. Kechris and W. Hugh Woodin; 18. A coding theorem for measures Alexander S. Kechris; 19. The tree of a Moschovakis scale is homogeneous Donald A. Martin and John R. Steel; 20. Weakly homogeneous trees Donald A. Martin and W. Hugh Woodin
Author(s): Alexander S. Kechris, Benedikt Lowe, John R. Steel
Series: Lecture Notes in Logic 31
Edition: 1
Year: 2008
Language: English
Pages: 460
COVER......Page 1
HALF-TITLE......Page 3
SERIES-TITLE......Page 5
TITLE......Page 7
COPYRIGHT......Page 8
CONTENTS......Page 9
PREFACE......Page 11
PART I: GAMES AND SCALES......Page 14
§1. Some definitions and history......Page 16
2.2. The tree to produce an elementary submodel......Page 19
2.3. Propagation of scales using comparison games......Page 20
Notes on the theory of scales [KM78B]......Page 23
Propagation of the scale property using games [Nee07]. Scales on…......Page 26
Inductive scales on inductive sets [Mos78]. Scales on coinductive sets [Mos83]. The extent of scales in L(R) [MS83]......Page 29
The largest countable this, that, and the other [Mar83A]......Page 31
Scales in K(R) [Ste07C]......Page 33
The length…......Page 34
REFERENCES......Page 37
NOTES ON THE THEORY OF SCALES......Page 41
§1. Preliminaries......Page 42
2.1. Definition and elementary properties......Page 43
2.2. Establishing the Prewellordering property......Page 44
2.3. The First Periodicity Theorem......Page 46
2.4. The zig-zag picture......Page 49
3.1. Definitions and basic properties......Page 50
3.2. Establishing the Scale property......Page 52
3.3. The Second Periodicity Theorem......Page 55
§4. Bases......Page 59
4.1. Computation of bases......Page 60
§5. Partially playful universes......Page 61
6.1. Notation for trees......Page 64
6.2. κ-scales and their trees......Page 65
6.3. Computing lengths of scales......Page 66
7.1. The Kunen-Martin theorem......Page 67
8.1. Inductive analysis of projection of trees......Page 69
2. An extension of Suslin’s Theorem......Page 71
8.3. Unions of Borel sets......Page 72
9.1. The models L…......Page 73
9.2. Absoluteness of closed games......Page 75
9.3. Proof that......Page 76
10.1. Sets which are ∞-Boolean over a model of ZFC......Page 79
10.2. Ideals of Borel sets which are suitable over a model of ZFC......Page 80
11.1. The theorem on perfect sets......Page 82
11.2. Largest countable......Page 83
§12. A summary of results about projective ordinals......Page 85
REFERENCES......Page 86
PROPAGATION OF THE SCALE PROPERTY USING GAMES......Page 88
REFERENCES......Page 102
SCALES ON…......Page 103
REFERENCES......Page 106
§1. Proof of the Main Theorem......Page 107
§2. Corollaries and remarks......Page 112
REFERENCES......Page 114
§1. Lemmas on preservation of scales......Page 115
§2. The main result......Page 119
REFERENCES......Page 122
THE EXTENT OF SCALES IN L(R)......Page 123
REFERENCES......Page 133
§1. Introduction......Page 134
§3. Characterization of C2i and of 0#......Page 139
§4. The largest countable inductive set......Page 140
REFERENCES......Page 142
§0. Introduction......Page 143
§1. The fine structure of L(R)......Page 144
§2. Scales on…......Page 153
§3. Scales at the end of a…......Page 162
§4. Suslin cardinals......Page 176
REFERENCES......Page 187
§1. Introduction......Page 189
2.1. Potential R-premice......Page 190
2.3. Ultrapowers, Iteration Trees......Page 193
§3. Some local HOD’s......Page 195
4.1. Scales on…......Page 206
4.2.…gaps.......Page 211
4.3. Scales at the end of a gap......Page 213
REFERENCES......Page 220
§1. Introduction......Page 222
§2. Infimum norms......Page 223
§3. Supremum norms......Page 225
§4. Game norms......Page 227
§5. The main construction......Page 229
§6. Canonical winning strategies......Page 232
§7. Definability......Page 233
REFERENCES......Page 235
LONG GAMES......Page 236
§1. Some game quantifierswhich propagate scales......Page 237
§2. Canonical strategies......Page 246
§3. An inner model of AD…......Page 255
§4. A determinacy proof......Page 264
§5. Questions......Page 270
REFERENCES......Page 272
§1. Introduction......Page 273
§2. The Prewellordering Property......Page 275
§3. The Scale Property......Page 276
REFERENCES......Page 282
PART II: SUSLIN CARDINALS, PARTITION PROPERTIES, HOMOGENEITY......Page 284
§1. Introduction......Page 286
§2. Suslin Cardinals......Page 292
§3. Partition Properties......Page 303
§4. Homogeneous Trees......Page 310
REFERENCES......Page 324
SUSLIN CARDINALS, kappa-SUSLIN SETS AND THE SCALE PROPERTY IN THE HYPERPROJECTIVE HIERARCHY......Page 327
§1. Suslin cardinals and Kappa-Suslin sets......Page 328
§2. Suslin cardinals below KappaR......Page 331
§3. The scale property in the hyperprojective hierarchy......Page 336
§4. Gaps in the propagation of scales......Page 340
5.1. Closure properties of S(kappa)......Page 341
5.5. Reliable cardinals......Page 343
REFERENCES......Page 344
THE AXIOM OF DETERMINACY, STRONG PARTITION PROPERTIES AND NONSINGULAR MEASURES......Page 346
§1. A partition theorem......Page 347
§2. Partition properties imply determinacy......Page 354
§3. Mahlo cardinals from determinacy......Page 362
§4. Nonsingular measures from AD......Page 364
REFERENCES......Page 366
1.1. Let L(R)......Page 368
§2. Proof of Theorem 1.8......Page 372
§3. Proof of Theorem 1.10......Page 380
§4. Proof of Theorem 1.2......Page 382
REFERENCES......Page 390
§1. A simple lemma......Page 392
§2. An ordinal determinacy result......Page 394
§3. Generic codes for ordinals......Page 396
§4. Some new partition properties......Page 398
§5. The weak Baire theory for omega Lambda......Page 400
§6. Generic elementary embeddings for the Levy collapse......Page 402
§7. Some applications of the generic elementary embeddings......Page 406
§8. Addendum; non-existence of strong codes......Page 407
§9. Addendum; Ordinal games and reliable cardinals......Page 409
REFERENCES......Page 410
§1. Introduction......Page 411
§2. A game for coding ultrafilters......Page 412
§3. On real-integer games......Page 413
§4. The complexity of ultrafilters......Page 414
§5. Absoluteness of ultrafilters......Page 415
REFERENCES......Page 416
§0......Page 417
§1......Page 418
REFERENCES......Page 433
§1. Introduction and background......Page 434
§2. The main lemmas......Page 437
§3. The main theorems......Page 441
§4. Further results and open problems......Page 444
REFERENCES......Page 450
BIBLIOGRAPHY......Page 452
Lecture Notes in Logic......Page 459
