Wadge Degrees and Projective Ordinals: The Cabal Seminar Volume II

Cover......Page 1
Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II......Page 2
LECTURE NOTES IN LOGIC......Page 4
Title......Page 6
Copyright......Page 7
CONTENTS......Page 8
PREFACE......Page 10
Volume I......Page 14
Volume II......Page 18
PART III: WADGE DEGREES AND POINTCLASSES......Page 24
§1. Introduction......Page 26
§2. Some basic facts about the Wadge hierarchy......Page 29
§3. The papers in the volume.......Page 32
§4. Recent developments......Page 38
REFERENCES......Page 43
§1. Definitions.......Page 47
§2. The Lipschitz ordering.......Page 48
§3. The Wadge Ordering......Page 51
§4. The Order Type of the Δ1
n Degrees.......Page 53
§5. Separation, Reduction, and Prewellordering Properties in the Wadge Hierarchy......Page 57
REFERENCES......Page 64
A NOTE ON WADGE DEGREES......Page 66
REFERENCES......Page 69
SOME RESULTS IN THE WADGE HIERARCHY OF BOREL SETS......Page 70
§1. A Description of Wadge Classes of Borel Sets.......Page 71
§2. Effective Results in the Borel Wadge hierarchy.......Page 87
REFERENCES......Page 95
THE STRENGTH OF BOREL WADGE DETERMINACY......Page 97
§1. Descriptions of Borel Wadge Classes.......Page 98
§2. Ramifications of Closed Games.......Page 104
§3. Proof of Borel Wadge Determinacy.......Page 110
§4. WadgeClasses inMetric Separable Spaces.......Page 115

§5. Hurewicz Tests and Hurewicz-Type Results.......Page 121
REFERENCES......Page 124
CLOSURE PROPERTIES OF POINTCLASSES......Page 125
§1. Consequences of the separation property......Page 126
§2. Applications of the Martin-Monk method......Page 128
§3. Bounded unions and prewellorderings
.......Page 134
REFERENCES......Page 139
§1. Introduction.......Page 141
§2. A criterion for PWO(Γ)......Page 142
2.1. Γ is closed under both ∃R and ∀R.......Page 143
2.2. Γ is closed under ∃R but not ∀R......Page 146
2.3. Γ is closed under ∀R but not ∃R.......Page 147
§3. Inductive-like pointclasses and projective algebras......Page 149
§4. Projective-like pointclasses and hierarchies.......Page 153
§5. The prewellordering pattern in projective-like hierarchies.......Page 156
§6. Problems and conjectures.......Page 159
Appendix A......Page 160
Appendix B......Page 161
REFERENCES......Page 162
§1. Introduction......Page 164
§2. Inductive-like pointclasses.......Page 166
§3. κ-Suslin sets for κ a successor cardinal.......Page 168
§4. κ-Suslin sets for κ a limit cardinal of uncountable cofinality.......Page 173
§5. Addendum (2010)......Page 174
REFERENCES......Page 175
MORE CLOSURE PROPERTIES OF POINTCLASSES......Page 177
REFERENCES......Page 181
MORE MEASURES FROM AD......Page 183
REFERENCES......Page 188
§1. Introduction......Page 189
§3. Descriptive set theory......Page 190
§5. The analogy with recursive function theory......Page 191
§6. The Baire Space.......Page 192
§7. Clopen sets as recursive sets.......Page 193
§9. Continuous functions as computable functions.......Page 194
§10. Luzin’s examples.......Page 195
§12. Many-one reducibility.......Page 196
§14. The gameG(A,B).......Page 197
§15. Completeness of Luzin’s sets.......Page 198
§16. The Δ0
2 degrees.......Page 199
§17. The determinacy of G(A,B).......Page 201
§18. The Axiom of Determinacy.......Page 202
§19. Degree arithmetic......Page 203
§20. (α, )-homeomorphisms......Page 204
§21. The expansion operations.......Page 205
§22. The ordinal jump functions......Page 206
§23. Boolean set operations.......Page 207
§24.
G-Boolean classes.......Page 208
§25. Separated and partitioned unions.......Page 209
§26. Determining ϑ1......Page 211
§27. The arithmetic degrees.......Page 212
§28. Luzin’s problem......Page 213

§29. Determining Ξ.......Page 214
§30. The Borel degrees.......Page 215
REFERENCES......Page 216
PART IV: PROJECTIVE ORDINALS......Page 220
§1. Introduction.......Page 222
§2. Background and Preliminaries.......Page 226
§3. Outline of the Arguments.......Page 235
§4. The First Level Theory.......Page 243
§5. The Second Level of the Induction.......Page 259
§6. Level-3 descriptions.......Page 271
§7. Higher Levels.......Page 283
§8. Concluding Remarks.......Page 288
REFERENCES......Page 291
0.1. Trees......Page 293
0.3. Indiscernibles.......Page 294
1.1. Definition of S1.......Page 295
1.2. Scales for Π1
sets.......Page 296
1.3. Homogeneity properties of S1.......Page 297
2.1. Definition of S2.......Page 298
2.3. Homogeneity properties of S2. The tree S−......Page 301
2.4. Some definability estimates for S−
2......Page 304
2.5. An alternative tree S2+.......Page 305
3.1. Definition of S3.......Page 307
3.2. Scales for Π1
3 sets.......Page 310
3.3. Homogeneity properties of S3. The tree S−
3 .......Page 311
4.1. Definition of S4.......Page 313
4.2. Scales for Π1
4 sets......Page 314
§6. Homogeneous trees in general.......Page 316
§7. A result of Martin on subsets of
1
.......Page 320
§8. On the Victoria Delfino Third Problem.......Page 322
REFERENCES......Page 325
AD AND PROJECTIVE ORDINALS......Page 327
§2. For all n, 1  n is a cardinal.......Page 328
§3. The 1
’s are successor cardinals.......Page 330
§4. The 1
’s are regular.......Page 336
§5. The 1
’s are measurable.......Page 337
§6. Calculating 1
n for n ≤ 4.......Page 339
§7. The closed unbounded measure on 1.......Page 341
§8. Uniform indiscernibles and the n’s for n ≤ .......Page 342
§9. Back to the real world.......Page 346
§10. Infinite exponent partition relations and the singular measures  .......Page 348
§11. Countable exponent partition relations for 1
n, n odd.......Page 349
§12. 1 → (1)1
.......Page 351
§13. The Martin-Paris theorem.......Page 352
§14. The measure  on 1
, n odd.......Page 356
§15. The measures , with >,on 1
n, n odd.......Page 359
§16. Countable exponent partition relations on 1
n, n even......Page 360
§17. The measure  on 1
n, n even.......Page 362
§18. Some singular cardinals.......Page 365
REFERENCES......Page 367
§1. Introduction.......Page 369
§2. Classification of tuples of ordinals.......Page 370
§3. Applications.......Page 383
REFERENCES......Page 386
§2. Definitions and preliminary results.......Page 387
§3. A Global Embedding Theorem......Page 399
§4. A Local Embedding Theorem.......Page 409
§5. The Main Lemma.......Page 426
5.1. Proof of themain inductive lemma.......Page 456
5.3. H2n+1(a).......Page 472
5.4. H2n+1(b).......Page 475
5.6. H2n+1(d).......Page 476
§6. The Main Theorem.......Page 477
§7. A Rank Computation.......Page 479
§9. A Lower Bound for fp.......Page 497
REFERENCES......Page 505
§1. Introduction.......Page 507
2.1.......Page 510
3.1. Finite sequences and trees.......Page 511
3.2.......Page 513
3.3.......Page 514
3.4.......Page 515
§4. Σ1
sets.......Page 516
5.1.1.......Page 518
5.1.2.......Page 519
5.2.1. The class of uniform indiscernibles is C =  Cx.......Page 520
5.2.2.......Page 521
5.3.1.......Page 522
§6. Higher levels in the projective hierarchy.......Page 525
§7. Consequences of the full axiom of determinacy.......Page 526
REFERENCES......Page 530
§2. Negative Partition Results.......Page 532
§3. Positive Partition Results......Page 536
REFERENCES......Page 539
BIBLIOGRAPHY......Page 542