Model Theory (Encyclopedia of Mathematics and its Applications, Series Number 42)

Professor Hodges emphasizes 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. Contents......Page 5 Introduction......Page 9 Note on notation......Page 12 1 Naming of parts......Page 15 1.1 Structures......Page 16 1.2 Homomorphisms and substructures......Page 19 1.3 Terms and atomic formulas......Page 25 1.4 Parameters and diagrams......Page 29 1.5 Canonical models......Page 32 History and bibliography......Page 35 2 Classifying structures......Page 37 2.1 Definable subsets......Page 38 2.2 Definable classes of structures......Page 47 2.3 Some notions from logic......Page 54 2.4 Maps and the formulas they preserve......Page 61 2.5 Classifying maps by formulas......Page 68 2.6 Translations......Page 72 2.7 Quantifier elimination......Page 80 2.8 Further examples......Page 89 History and bibliography......Page 96 3.1 Theorems of Skolem......Page 101 3.2 Back-and-forth equivalence......Page 108 3.3 Games for elementary equivalence......Page 116 3.4 Closed games......Page 125 3.5 Games and infinitary languages......Page 133 3.6 Clubs......Page 138 History and bibliography......Page 142 4 Automorphisms......Page 145 4.1 Automorphisms......Page 146 4.2 Subgroups of small index......Page 154 4.3 Imaginary elements......Page 162 4.4 Eliminating imaginaries......Page 171 4.5 Minimal sets......Page 177 4.6 Geometries......Page 184 4.7 Almost strongly minimal theories......Page 192 4.8 Zil'ber's configuration......Page 202 History and bibliography......Page 211 5 Interpretations......Page 215 5.1 Relativisation......Page 216 5.2 Pseudo-elementary classes......Page 220 5.3 Interpreting one structure in another......Page 226 5.4 Shapes and sizes of interpretations......Page 233 5.5 Theories that interpret anything......Page 241 5.6 Totally transcendental structures......Page 251 5.7 Interpreting groups and fields......Page 262 History and bibliography......Page 274 6 The first-order case: compactness......Page 278 6.1 Compactness for first-order logic......Page 279 6.2 Boolean algebras and Stone spaces......Page 285 6.3 Types......Page 291 6.4 Elementary amalgamation......Page 299 6.5 Amalgamation and preservation......Page 308 6.6 Expanding the language......Page 314 6.7 Stability......Page 320 History and bibliography......Page 332 7.1 Fraisse's construction......Page 337 7.2 Omitting types......Page 347 7.3 Countable categoricity......Page 355 7.4 cocategorical structures by Fraisse's method......Page 362 History and bibliography......Page 371 8 The existential case......Page 374 8.1 Existentially closed structures......Page 375 8.2 Two methods of construction......Page 380 8.3 Model-completeness......Page 388 8.4 Quantifier elimination revisited......Page 395 8.5 More on e.c. models......Page 405 8.6 Amalgamation revisited......Page 414 History and bibliography......Page 423 9 The Horn case: products......Page 426 9.1 Direct products......Page 427 9.2 Presentations......Page 434 9.3 Word-constructions......Page 444 9.4 Reduced products......Page 455 9.5 Ultraproducts......Page 463 9.6 The Feferman-Vaught theorem......Page 472 9.7 Boolean powers......Page 480 History and bibliography......Page 487 10 Saturation......Page 492 10.1 The great and the good......Page 493 10.2 Big models exist......Page 503 10.3 Syntactic characterisations......Page 510 10.4 Special models......Page 520 10.5 Definability......Page 529 10.6 Resplendence......Page 536 10.7 Atomic compactness......Page 541 History and bibliography......Page 546 11 Combinatorics......Page 549 11.1 Indiscernibles......Page 550 11.2 Ehrenfeucht-Mostowski models......Page 559 11.3 EM models of unstable theories......Page 569 11.4 Nonstandard methods......Page 581 11.5 Defining well-orderings......Page 590 11.6 Infinitary indiscernibles......Page 600 History and bibliography......Page 608 12 Expansions and categoricity......Page 613 12.1 One-cardinal and two-cardinal theorems......Page 614 12.2 Categoricity......Page 625 12.3 Cohomology of expansions......Page 638 12.4 Counting expansions......Page 646 12.5 Relative categoricity......Page 652 History and bibliography......Page 663 A.1 Modules......Page 667 A.2 Abelian groups......Page 676 A.3 Nilpotent groups of class 2......Page 687 A.4 Groups......Page 702 A.5 Fields......Page 709 A.6 Linear orderings......Page 720 References......Page 730 Index to symbols......Page 769 Index......Page 771 "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."--Page [4] of cover "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."--Jacket

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.