A guide to classical and modern model theory
著者
書誌事項
A guide to classical and modern model theory
（Trends in logic : studia logica library, v. 19）
Kluwer Academic, c2003
- : pbk
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注記
Includes bibliographical references and index
内容説明・目次
- 巻冊次
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ISBN 9781402013300
内容説明
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 fertile research area which should be of interest to the mathematical world. This volume should be easily accessible to young people and mathematicians unfamiliar with logic; it gives a terse historical picture of Model Theory; it introduces the latest developments in the area; and it provides "hands-on" proofs of elimination of quantifiers, elimination of imaginaries and other relevant matters. The text is for trainees and professional model theorists, mathematicians working in algebra and geometry and young people with a basic knowledge of logic.
- 巻冊次
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: pbk ISBN 9781402013317
内容説明
This volume is easily accessible to young people and mathematicians unfamiliar with logic. It gives a terse historical picture of Model Theory and introduces the latest developments in the area. It further provides 'hands-on' proofs of elimination of quantifiers, elimination of imaginaries and other relevant matters. The book is for trainees and professional model theorists, and mathematicians working in Algebra and Geometry.
目次
1: 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.
2: 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.
3: Model Completeness. 3.1. An introduction. 3.2. Abraham Robinson's test. 3.3. Model completeness and algebra. 3.4. p-adic fields and Artin's conjecture. 3.5. Existentially closed fields. 3.6. DCF0. 3.7. SCFp and DCFp. 3.8. ACFA. 3.9. References.
4: 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.
5: 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 models. 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.
6: Omega-stability. 6.1. Totally transcendental theories. 6.2. omega-stable groups. 6.3. omega-stable fields. 6.4. Prime models. 6.5. DCF0 revisited. 6.6. Ryll-Nardzewski's Theorem and other things. 6.7. References.
7: Classifying. 7.1. Shelah's Classification Theory. 7.2. Simple theories. 7.3. Stable theories. 7.4. Superstable theories. 7.5. w-stable theories. 7.6. Classifiable theories. 7.7. Shelah's Uniqueness Theorem. 7.8. Morley's Theorem. 7.9. Biinterpretability and Zilber Conjecture. 7.10. Two algebraic examples. 7.11. References.
8: 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.
9: 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.
Bibliography. Index.
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