COURSE ON MODEL THEORY

Introduction 2 0.1. Some theorems in mathematics with snappy model-theoretic proofs 2 1. Languages, structures, sentences and theories 2 1.1. Languages 2 1.2. Statements and Formulas 5 1.3. Satisfaction 6 1.4. Elementary equivalence 7 1.5. Theories 8 2. Big Theorems: Completeness, Compactness and Löwenheim-Skolem 9 2.1. The Completeness Theorem 9 2.2. Proof-theoretic consequences of the completeness theorem 11 2.3. The Compactness Theorem 13 2.4. Topological interpretation of the compactness theorem 13 2.5. First applications of compactness 15 2.6. The Löwenheim-Skolem Theorems 17 3. Complete and model complete theories 19 3.1. Maximal and complete theories 19 3.2. Model complete theories 20 3.3. Algebraically closed fields I: model completeness 21 3.4. Algebraically closed fields II: Nullstellensätze 22 3.5. Algebraically closed fields III: Ax’s Transfer Principle 24 3.6. Ordered fields and formally real fields I: background 25 3.7. Ordered fields and formally real fields II: the real spectrum 26 3.8. Real-closed fields I: definition and model completeness 26 3.9. Real-closed fields II: Nullstellensatz 27 3.10. Real-closed fields III: Hilbert’s 17th problem 30 4. Categoricity: a condition for completeness 30 4.1. DLO 32 4.2. R-modules 33 4.3. Morley’s Categoricity Theorem 35 4.4. Complete, non-categorical theories 35 5. Quantifier elimination: a criterion for model-completeness 36 5.1. Constructible and definable sets 36 5.2. Quantifier Elimination: Definition and Implications 39 5.3. A criterion for quantifier elimination 41 5.4. Model-completeness of ACF 43 5.5. Model-completeness of RC(O)F 44 5.6. Algebraically Prime Models 45

[1]  Guy Terjanian.  Formes p-adiques anisotropes. , 1980 .

[2]  Julia Robinson,et al.  Definability and decision problems in arithmetic , 1949, Journal of Symbolic Logic.

[3]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[4]  Wilfrid Hodges,et al.  A Shorter Model Theory , 1997 .

[5]  A. Robinson.  On ordered fields and definite functions , 1955 .

[6]  Abraham Robinson,et al.  Solution of an invariant subspace problem of K , 1966 .

[7]  B. Poizat.  A Course in Model Theory , 2000 .

[8]  D. Hoffmann,et al.  INTRODUCTION TO QUADRATIC FORMS OVER FIELDS (Graduate Studies in Mathematics 67) By T. Y. L AM : 550 pp., US$79.00, ISBN 0-8218-1095-2 (American Mathematical Society, Providence, RI, 2005) , 2005 .

[9]  S. Lang.  On Quasi Algebraic Closure , 1952 .

[10]  John L. Bell,et al.  Models and Ultraproducts: An Introduction. , 1969 .

[11]  Moshe Jarden,et al.  THE ELEMENTARY THEORY OF FINITE FIELDS , 2004 .

[12]  Walter Rudin,et al.  Injective Polynomial Maps Are Automorphisms , 1995 .

[13]  S. Kochen,et al.  Diophantine Problems Over Local Fields I , 1965 .

[14]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[15]  Jean-Pierre Serre,et al.  How to use finite fields for problems concerning infinite fields , 2009, 0903.0517.

[16]  H. Jerome Keisler,et al.  Model theory, Third Edition , 1992, Studies in logic and the foundations of mathematics.

[17]  Tsit Yuen Lam,et al.  Introduction To Quadratic Forms Over Fields , 2004 .

[18]  Armand Borel,et al.  Injective endomorphisms of algebraic varieties , 1969 .

[19]  Perry Smith.  Review: J. L. Bell, A. B. Slomson, Models and Ultraproducts: An Introduction , 1972 .