A Guide to Classical and Modern Model Theory

ISBN-13: 9781402013317 / Angielski / Miękka / 2003 / 371 str.

Annalisa Marcja; Carlo Toffalori

Opis

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.

Spis treści

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.