.

0 (hodnocen0 x )

BK

1st ed.

Cambridge : Cambridge University, 1995

xiii,772 s.

ISBN 0-521-30442-3 (váz.)

Encyclopedia of mathematics and its applications ; [vol.] 42

Obsahuje úvod, poznámky, rejstříky

Bibliografie: s. 716-754

000062023

Paperback Re-Issue // This is an up-to-date and integrated introduction to model theory, designed to be used for graduate courses (for students who are familiar with first-order logic), and as a reference for more experienced logicians and mathematicians. // Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide variety of other areas such as set theory, geometry, algebra (in particular group theory), and computer science (e.g. logic programming and specification). Professor Hodges emphasises definability and methods of construction, and introduces the reader to advanced topics such as stability. He also provides the reader with much historical information and a full bibliography, enhancing the book’s use as a reference. // Cambrid // UNIVERSITY PR www.cambridge.org // ISBN 0-521-06636 // 9 78052 06636 // Introduction ix // Note on notation xii // 1 Naming of parts 1 // 1.1 Structures 2 // 1.2 Homomorphisms and substructures 5 // 1.3 Terms and atomic formulas 11 // 1.4 Parameters and diagrams 15 // 1.5 Canonical models 18 // History and bibliography 21 // 2 Classifying structures 23 // 2.1 Definable subsets 24 // 2.2 Definable classes of structures 33 // 2.3 Some notions from logic 40 // 2.4 Maps and the formulas they preserve 47 // 2.5 Classifying maps by formulas 54 // 2.6 Translations 58 // 2.7 Quantifier elimination 66 // 2.8 Further examples 75 // History and bibliography 82 // 3 Structures

that look alike 87 // 3.1 Theorems of Skolem 87 // 3.2 Back-and-forth equivalence 94 // 3.3 Games for elementary equivalence 102 // 3.4 Closed games 111 // 3.5 Games and infinitary languages 119 // 3.6 Clubs 124 // History and bibliography 128 // vi Contents // 4 Automorphisms 131 // 4.1 Automorphisms 132 // 4.2 Subgroups of small index 140 // 4.3 Imaginary elements 148 // 4.4 Eliminating imaginaries 157 // 4.5 Minimal sets 163 // 4.6 Geometries 170 // 4.7 Almost strongly minimal theories 178 // 4.8 Zilber’s configuration 188 // History and bibliography 197 // 5 Interpretations 201 // 5.1 Relati visation 202 // 5.2 Pseudo-elementary classes 206 // 5.3 Interpreting one structure in another 212 // 5.4 Shapes and sizes of interpretations 219 // 5.5 Theories that interpret anything 227 // 5.6 Totally transcendental structures 237 // 5.7 Interpreting groups and fields 248 // History and bibliography 260 // 6 The first-order case: compactness 264 // 6.1 Compactness for first-order logic 265 // 6.2 Boolean algebras and Stone spaces 271 // 6.3 Types 277 // 6.4 Elementary amalgamation 285 // 6.5 Amalgamation and preservation 294 // 6.6 Expanding the language 300 // 6.7 Stability 306 // History and bibliography 318 // 7 The countable case 323 // 7.1 Fraě’ssé’s construction 323 // 7.2 Omitting types 333 // 7.3 Countable categoricity 341 // 7.4 co-categorical structures by Fraě’ssé’s method 348 // History and bibliography 357 // 8 The existential case 360 // 8.1 Existentially closed

structures 361 // 8.2 Two methods of construction 366 // 8.3 Model-completeness 374 // 8.4 Quantifier elimination revisited 381 // 8.5 More on e.c. models 391 // 8.6 Amalgamation revisited 400 // History and bibliography 409 // Contents // vii // 9 The Horn case: products 412 // 9.1 Direct products 413 // 9.2 Presentations 420 // 9.3 Word-constructions 430 // 9.4 Reduced products 441 // 9.5 Ultraproducts 449 // 9.6 The Feferman-Vaught theorem 458 // 9.7 Boolean powers 466 // History and bibliography 473 // 10 Saturation 478 // 10.1 The great and the good 479 // 10.2 Big models exist 489 // 10.3 Syntactic characterisations 496 // 10.4 Special models 506 // 10.5 Definability 515 // 10.6 Resplendence 522 // 10.7 Atomic compactness 527 // History and bibliography 532 // 11 Combinatorics 535 // 11.1 Indiscernibles 536 // 11.2 Ehrenfeucht-Mostowski models 545 // 11.3 EM models of unstable theories 555 // 11.4 Nonstandard methods 567 // 11.5 Defining well-orderings 576 // 11.6 Infinitary indiscernibles 586 // History and bibliography 594 // 12 Expansions and categoricity 599 // 12.1 One-cardinal and two-cardinal theorems 600 // 12.2 Categoricity 611 // 12.3 Cohomology of expansions 624 // 12.4 Counting expansions 632 // 12.5 Relative categoricity 638 // History and bibliography 649 // Appendix: Examples 653 // A.l Modules 653 // A.2 Abelian groups 662 // A.3 Nilpotent groups of class 2 673 // A.4 Groups 688 // A.5 Fields 695 // A.6 Linear orderings 706 // 716 // 755 // References Index

to symbols Index // 757