This book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and Ramsey theory. In the second part, the sophisticated technique of forcing, originally developed by Paul Cohen, is explained in great detail. With this technique, one can show that certain statements, like the continuum hypothesis, are neither provable nor disprovable from the axioms of set theory. In the last part, some topics of classical set theory are revisited and further developed in the light of forcing. The notes at the end of each chapter put the results in a historical context, and the numerous related results and the extensive list of references lead the reader to the frontier of research. This book will appeal to all mathematicians interested in the foundations of mathematics, but will be of particular use to graduates in this field.
Author(s): Lorenz J. Halbeisen
Series: Springer Monographs in Mathematics
Edition: 2012
Publisher: Springer
Year: 2011
Language: English
Pages: 448
Preface......Page 5
Contents......Page 8
1. The Setting......Page 14
Notes......Page 18
References......Page 19
Part I. Topics in Combinatorial Set Theory......Page 20
The Nucleus of Ramsey Theory......Page 21
Corollaries of Ramsey's Theorem......Page 24
Generalisations of Ramsey's Theorem......Page 26
Finite Colourings of [omega]^{Going to the Infinite......Page 29
Notes......Page 30
Related Results......Page 31
References......Page 35
Why Axioms?......Page 37
Syntax: Formulae, Formal Proofs, and Consistency......Page 39
Semantics: Models, Completeness, and Independence......Page 46
Limits of First-Order Logic......Page 49
The Axioms of Zermelo-Fraenkel Set Theory......Page 51
Models of ZF......Page 64
Cardinals in ZF......Page 66
Notes......Page 69
Related Results......Page 76
References......Page 77
Basic Cardinal Relations......Page 83
On the Cardinals 20 and 1......Page 88
Ordinal Numbers Revisited......Page 91
fin(m)<2^m whenever m is infinite......Page 95
seq^{1-1}(m)<>2^m<>seq(m) whenever m>=2......Page 98
2^{2^m} + 2^{2^m} = 2^{2^m} whenever m is infinite......Page 103
Notes......Page 107
Related Results......Page 108
References......Page 111
Zermelo's Axiom of Choice and Its Consistency with ZF......Page 113
Equivalent Forms of the Axiom of Choice......Page 114
Cardinal Arithmetic in the Presence of AC......Page 123
The Prime Ideal Theorem and Related Statements......Page 128
König's Lemma and Other Choice Principles......Page 135
Notes......Page 138
Related Results......Page 143
References......Page 148
Equidecomposability......Page 154
Hausdorff's Paradox......Page 155
Robinson's Decomposition......Page 158
Notes......Page 163
Related Results......Page 164
References......Page 165
Permutation Models......Page 167
The Basic Fraenkel Model......Page 170
The Second Fraenkel Model......Page 172
The Ordered Mostowski Model......Page 174
The Prime Ideal Theorem Revisited......Page 177
Custom-Built Permutation Models......Page 179
The First Custom-Built Permutation Model......Page 180
The Second Custom-Built Permutation Model......Page 181
Notes......Page 183
Related Results......Page 184
References......Page 186
8. Twelve Cardinals and Their Relations......Page 188
The Cardinal p......Page 189
The Cardinals b and d......Page 190
The Cardinals s and r......Page 191
The Cardinals a and i......Page 193
The Cardinals par and hom......Page 197
The Cardinal h......Page 199
Summary......Page 201
Notes......Page 202
Related Results......Page 203
References......Page 206
The Ramsey Property......Page 209
The Ideal of Ramsey-Null Sets......Page 211
The Ellentuck Topology......Page 212
A Generalised Suslin Operation......Page 216
Related Results......Page 219
References......Page 220
Happy Families......Page 222
Ramsey Ultrafilters......Page 226
P-points and Q-points......Page 228
Ramsey Families and P-families......Page 232
Related Results......Page 237
References......Page 240
The Hales-Jewett Theorem......Page 241
Families of Partitions......Page 245
Carlson's Lemma and the Partition Ramsey Theorem......Page 248
A Weak Form of the Halpern-Läuchli Theorem......Page 255
Notes......Page 256
Related Results......Page 257
References......Page 260
Part II. From Martin's Axiom to Cohen's Forcing......Page 262
An example from Group Theory:......Page 263
An example from Peano Arithmetic:......Page 264
Filters on Partially Ordered Sets......Page 266
Weaker Forms of MA......Page 269
Some Consequences of MA(σ-centred)......Page 270
MA(countable) Implies the Existence of Ramsey Ultrafilters......Page 272
Notes......Page 273
References......Page 274
The Notion of Forcing Notion.......Page 276
Making Sets from Names.......Page 278
A Saucerful of Names.......Page 279
Generic Extensions......Page 280
Equivalent Forcing Notions.......Page 281
Alternative Definitions of Generic Filters.......Page 282
The Forcing Relationship.......Page 283
The Forcing Theorem.......Page 284
The Generic Model Theorem.......Page 288
Forcing Notions Which Do not Add Reals.......Page 289
Forcing Notions Which Do not Collapse Cardinals.......Page 291
Independence of CH: The Gentle Way......Page 292
On the Existence of Generic Filters......Page 294
Notes......Page 295
References......Page 296
Basic Model-Theoretical Facts......Page 297
The Reflection Principle......Page 298
Countable Transitive Models of Finite Fragments of ZFC......Page 301
Related Results......Page 304
References......Page 305
Consistency and Independence Proofs: The Proper Way......Page 306
The Cardinality of the Continuum......Page 309
Notes......Page 310
References......Page 311
Symmetric Submodels of Generic Extensions......Page 312
A Model in Which the Reals Cannot Be Well-Ordered......Page 314
A Model in Which Every Ultrafilter over ω Is Principal......Page 317
A Model with a Paradoxical Decomposition of the Real Line......Page 318
Simulating Permutation Models by Symmetric Models......Page 320
Notes......Page 325
Related Results......Page 326
References......Page 327
General Products of Forcing Notions......Page 328
Products of Cohen Forcing......Page 330
A Model in Which aTwo-Step Iterations......Page 334
General Iterations......Page 338
A Model in Which iRelated Results......Page 344
References......Page 346
A Model in Which p=c=ω_2......Page 347
On the Consistency of MA+¬CH......Page 349
p=c Is Preserved Under Adding a Cohen Real......Page 350
Related Results......Page 353
References......Page 354
Part III. Combinatorics of Forcing Extensions......Page 355
Dominating, Splitting, Bounded, and Unbounded Reals......Page 356
The Laver Property and Not Adding Cohen Reals......Page 358
The Notion of Properness......Page 359
Preservation Theorems for Proper Forcing Notions......Page 361
Related Results......Page 362
References......Page 363
Cohen Forcing Adds Unbounded but no Dominating Reals......Page 364
Cohen Reals and the Covering Number of Meagre Sets......Page 365
A Model in Which aA Model in Which s=bRelated Results......Page 372
References......Page 374
22. Silver-Like Forcing Notions......Page 375
Silver-Like Forcing Is Proper and ω^ω-Bounding......Page 376
A Model in Which dRelated Results......Page 378
References......Page 379
23. Miller Forcing......Page 381
Miller Forcing Is Proper and Adds Unbounded Reals......Page 382
Miller Forcing Does not Add Splitting Reals......Page 383
Miller Forcing Preserves P-Points......Page 386
A Model in Which rRelated Results......Page 389
References......Page 391
Mathias Forcing Adds Dominating Reals......Page 393
Mathias Forcing Is Proper and Has the Laver Property......Page 394
A Model in Which pRelated Results......Page 400
References......Page 401
There May Be a Ramsey Ultrafilter and cov(M)There May Be no Ramsey Ultrafilter and h=c......Page 404
Related Results......Page 413
References......Page 415
A Dual Form of Mathias Forcing......Page 416
A Dual Form of Ramsey Ultrafilters......Page 422
Related Results......Page 425
References......Page 427
Prelude......Page 428
Allemande......Page 429
Courante......Page 430
Sarabande......Page 431
Gavotte I & II......Page 432
References......Page 433
Index of Symbols......Page 435
Index of Names......Page 438
Index......Page 442
