In this research monograph, the author's work on classification and related topics are presented. This revised edition brings the book up to date with the addition of four new chapters as well as various corrections to the 1978 text.The additional chapters X - XIII present the solution to countable first order T of what the author sees as the main test of the theory. In Chapter X the Dimensional Order Property is introduced and it is shown to be a meaningful dividing line for superstable theories. In Chapter XI there is a proof of the decomposition theorems. Chapter XII is the crux of the matter: there is proof that the negation of the assumption used in Chapter XI implies that in models of T a relation can be defined which orders a large subset of m|M|. This theorem is also the subject of Chapter XIII.
Author(s): S. Shelah
Series: Studies in Logic and the Foundations of Mathematics
Edition: 2
Publisher: North Holland
Year: 1990
Language: English
Commentary: page 99 missing
Pages: 740
Contents......Page 6
Introduction......Page 12
Introduction to the Revised Edition......Page 16
Open Problems......Page 18
Added in Proof......Page 24
Notation......Page 32
1. Preliminaries and saturation......Page 36
2. Order, stability and indiscernables......Page 44
0. Introduction......Page 53
1. Ranks of types......Page 56
2. Stability, ranks and definability......Page 64
3. Ranks, degrees and superstability......Page 76
4. The f.c.p., the independence property and the strict order property......Page 97
0. Introduction......Page 117
1. Forking......Page 119
2. The finite equivalence relation theorem......Page 129
3. The instability spectrum......Page 135
4. Further properties of forking......Page 142
5. The first stability cardinal......Page 156
6. Imaginary elements......Page 164
7. Instability......Page 171
0. Introduction......Page 184
1. The set of axioms......Page 186
2. Examples of F's......Page 191
3. General properties of F-primary models......Page 208
4. Prime models for stable theories......Page 217
5. Various results......Page 238
0. Introduction......Page 257
1. Orthogonality, regularity and minimality of types......Page 264
2. Dimensions and orders between indiscernable sets......Page 274
3. Weighted dimensions and superstability......Page 283
4. Semi-regular and semi-minimal types......Page 301
5. Multi-dimensional theories......Page 318
6. Cardinality-quantifiers and two-cardinal theorems......Page 323
7. Ranks revisited......Page 339
0. Introduction......Page 355
1. Reduced products and regular filters......Page 358
2. Good filters and compactness of reduced products......Page 367
3. Constructing ultrafilters......Page 379
4. Keisler's order......Page 404
5. Saturation of ultrapowers and categoricity of pseudo-elementary classes......Page 413
6. Saturation of ultralimits......Page 424
0. Introduction......Page 431
1. Skolem functions and generalizations of saturativity......Page 434
2. Generalized Ehrenfeucht-Mostowski models......Page 445
3. On the f.c.p., uniform trees, and |D(T)|>|T|=א_0......Page 453
4. Semi-definability......Page 460
5. Hanf numbers of omitting types......Page 466
0. Introduction......Page 474
1. Independance of types......Page 478
2. Unsuperstable theories......Page 489
3. Saturated models and the case λ=|T_1|......Page 498
4. Categoricity, saturation and homegeneity up to a cardinality......Page 505
0. Introduction......Page 513
1. Superstable theories and categoricity......Page 515
2. On the lower parts of the spectrum......Page 531
0. Introduction......Page 542
1. Preliminaries......Page 543
2. The dimensional order property......Page 546
3. The decomposition lemma......Page 554
4. Deepness......Page 561
5. Deep theories have many non-isomorphic models......Page 567
6. Infinite depth......Page 582
7. Trivial types......Page 584
1. Stationarization......Page 591
2. The axiomatic treatment......Page 595
3. Specifying the axiomatic treatment......Page 606
0. Introduction......Page 624
1. On F^k_λ and F^j_λ......Page 625
2. Stable systems......Page 632
3. On good sets......Page 637
4. The otop / existence dichotomy......Page 642
5. From the (א_0,2)-existence property to the (λ,2)-existence property......Page 650
6. The book's main theorem......Page 654
0. Introduction......Page 656
1. Can the models by characterized by invariants......Page 657
2. On having many models, no one elementarily embeddable into another......Page 661
3. On the Morely conjecture......Page 668
4. I(א_α,T) for α large enough......Page 677
1. Filters, stationary sets and families of sets......Page 687
2. Partition theorems......Page 693
3. Various results......Page 700
Historical Remarks......Page 707
References......Page 718
Index of Definitions and Abbreviations......Page 725
Index of Symbols......Page 737
