Author	Marcja, Annalisa. author
Title	A Guide to Classical and Modern Model Theory [electronic resource] / by Annalisa Marcja, Carlo Toffalori
Imprint	Dordrecht : Springer Netherlands : Imprint: Springer, 2003
Connect to	http://dx.doi.org/10.1007/978-94-007-0812-9
Descript	XI, 371 p. online resource

Since its birth, Model Theory has been developing a number of methods and concepts that have their intrinsic relevance, but also provide fruitful and notable applications in various fields of Mathematics. It is a lively and fertile research area which deserves the attention of the mathematical world. This volume: -is easily accessible to young people and mathematicians unfamiliar with logic; -gives a terse historical picture of Model Theory; -introduces the latest developments in the area; -provides 'hands-on' proofs of elimination of quantifiers, elimination of imaginaries and other relevant matters. A Guide to Classical and Modern Model Theory is for trainees and professional model theorists, mathematicians working in Algebra and Geometry and young people with a basic knowledge of logic

Structures -- 1.1 Structures -- 1.2 Sentences -- 1.3 Embeddings -- 1.4 The Compactness Theorem -- 1.5 Elementary classes and theories -- 1.6 Complete theories -- 1.7 Definable sets -- 1.8 References -- Quantifier Elimination -- 2.1 Elimination sets -- 2.2 Discrete linear orders -- 2.3 Dense linear orders -- 2.4 Algebraically closed fields (and Tarski) -- 2.5 Tarski again: Real closed fields -- 2.6 pp-elimination of quantifiers and modules -- 2.7 Strongly minimal theories -- 2.8 o-minimal theories -- 2.9 Computational aspects of q. e -- 2.10 References -- Model Completeness -- 3.1 An introduction -- 3.2 Abraham Robinsonโ{128}{153}s test -- 3.3 Model completeness and Algebra -- 3.4 p-adic fields and Artinโ{128}{153}s Conjecture -- 3.5 Existentially closed structures -- 3.6 DCF0 -- 3.7 SCFp and DCFp -- 3.8 ACFA -- 3.9 References -- Elimination of imaginaries -- 4.1 Interpretability -- 4.2 Imaginary elements -- 4.3 Algebraically closed fields -- 4.4 Real closed fields -- 4.5 The elimination of imaginaries sometimes fails -- 4.6 References -- Morley rank -- 5.1 A tale of two chapters -- 5.2 Definable sets -- 5.3 Types -- 5.4 Saturated models -- 5.5 A parenthesis: pure injective modules -- 5.6 Omitting types -- 5.7 The Morley rank, at last -- 5.8 Strongly minimal sets -- 5.9 Algebraic closure and definable closure -- 5.10 References -- ? -stability -- 6.1 Totally transcendental theories -- 6.2 ?-stable groups -- 6.3 ?-stable fields -- 6.4 Prime models -- 6.5 DCF0 revisited -- 6.6 Ryll-Nardzewskiโ{128}{153}s Theorem, and other things -- 6.7 References -- Classifying -- 7.1 Shelahโ{128}{153}s Classification Theory -- 7.2 Simple theories -- 7.3 Stable theories -- 7.4 Superstable theories -- 7.5 ?-stable theories -- 7.6 Classifiable theories -- 7.7 Shelahโ{128}{153}s Uniqueness Theorem -- 7.8 Morleyโ{128}{153}s Theorem -- 7.9 Biinterpretability and Zilber Conjecture -- 7.10 Two algebraic examples -- 7.11 References -- Model Theory and Algebraic Geometry -- 8.1 Introduction -- 8.2 Algebraic varieties, ideals, types -- 8.3 Dimension and Morley rank -- 8.4 Morphisms and definable functions -- 8.5 Manifolds -- 8.6 Algebraic groups -- 8.7 The Mordell-Lang Conjecture -- 8.8 References -- O-minimality -- 9.1 Introduction -- 9.2 The Monotonicity Theorem -- 9.3 Cells -- 9.4 Cell decomposition and other theorems -- 9.5 Their proofs -- 9.6 Definable groups in o-minimal structures -- 9.7 O-minimality and Real Analysis -- 9.8 Variants on the o-minimal theme -- 9.9 No rose without thorns -- 9.10 References