Model-Theoretic Logics

Description

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the eighth publication in the Perspectives in Logic series, brings together several directions of work in model theory between the late 1950s and early 1980s. It contains expository papers by pre-eminent researchers. Part I provides an introduction to the subject as a whole, as well as to the basic theory and examples. The rest of the book addresses finitary languages with additional quantifiers, infinitary languages, second-order logic, logics of topology and analysis, and advanced topics in abstract model theory. Many chapters can be read independently. Contents List of Contributors Part A. Introduction, Basic Theory and Examples Chapter I. Model-Theoretic Logics: Background and Aims 1. Logics Embodying Mathematical Concepts 2. Abstract Model Theory 3. Conclusion Chapter II. Extended Logics: The General Framework 1. General Logics 2. Examples of Principal Logics 3. Comparing Logics 4. Lindstrom Quantifiers 5. Compactness and Its Neighbourhood 6. Lowenheim-Skolem Properties 7. Interpolation and Definability Chapter III. Characterizing Logics 1. Lindstrom's Characterizations of First-Order Logic 2. Further Characterizations of ℒωω 3. Characterizing ℒ∞ω 4. Characterizing Cardinality Quantifiers 5. A Lindstrom-Type Theorem for Invariant Sentences Part B. Finitary Languages with Additional Quantifiers Chapter IV. The Quantifier "There Exist Uncountably Many" and Some of Its Relatives 1. Introduction to ℒ(Qα) 2. A Framework for Reducing to First-Order Logic 3. ℒ(Q1) and ℒω1ω(Q1): Completeness and Omitting Types Theorems 4. Filter Quantifiers Stronger Than Q1: Completeness, Compactness, and Omitting Types 5. Extensions of if ℒ(Q1) by Quantifiers Asserting the Existence of Certain Uncountable Sets 6. Interpolation and Preservation Questions 7. Appendix (An Elaboration of Section 2) Chapter V. Transfer Theorems and Their Applications to Logics 1. The Notions of Transfer and Reduction 2. The Classical Transfer Theorems 3. Two-Cardinal Theorems and the Method of Identities 4. Singular Cardinal-like Structures 5. Regular Cardinal-like Structures 6. Self-extending Models 7. Final Remarks Chapter VI. Other Quantifiers: An Overview 1. Quantifiers from Partially Ordered Prefixes 2. Quantifiers for Comparing Structures 3. Cardinality, Equivalence, Order Quantifiers and All That 4. Quantifiers from Robinson Equivalence Relations Chapter VII. Decidability and Quantifier-Elimination 1. Quantifier-Elimination 2. Interpretations 3. Dense Systems Part C. Infinitary Languages Chapter VIII. ℒω1ω and Admissible Fragments PART I. COMPACTNESS LOST 1. Introduction to Infinitary Logics 2. Elementary Equivalence 3. General Model-Theoretic Properties 4. "Harder" Model Theory PART II. COMPACTNESS REGAINED 5. Admissibility 6. General Model-Theoretic Properties with Admissibility 7. "Harder" Model Theory with Admissibility 8. Extensions of ifWlC0 by Propositional Connectives Appendix Chapter IX. Larger Infinitary Languages 1. The Infinitary Languages ℒkλ and ℒ∞λ 2. Basic Model Theory: Counterexamples 3. Basic Model Theory: The Lowenheim-Skolem Theorems 4. The Back-and-Forth Method Chapter X. Game Quantification 1. Infinite Strings of Quantifiers 2. Projective Classes and the Approximations of the Game Formulas 3. Model Theory for Game Logics 4. Game Quantification and Local Definability Theory Chapter XI Applications to Algebra 1. Universal Locally Finite Groups 2. Subdirectly Irreducible Algebras 3. Lefschetz's Principle 4. Abelian Groups 5. Almost-Free Algebras 6. Concrete Algebraic Constructions 7. Miscellany PartD. Second-Order Logic Chapter XII. Definable Second-Order Quantifiers 1. Definable Second-Order Quantifiers 2. Only Four Second-Order Quantifiers 3. Infinitary Monadic Logic and Generalized Products 4. The Comparison of Theories 5. The Classification of Theories by Interpretation of Second-Order Quantifiers 6. Generalizations Chapter XIII. Monadic Second-Order Theories 1. Monadic Quantification 2. The Automata and Games Decidability Technique 3. The Model-Theoretic Decidability Technique 4. The Undecidability Technique 5. Historical Remarks and Further Results Part E. Logics of Topology and Analysis Chapter XIV. Probability Quantifiers 1. Logic with Probability Quantifiers 2. Completeness Theorems 3. Model Theory 4. Logic with Conditional Expectation Operators 5. Open Questions and Research Problems Chapter XV. Topological Model Theory 1. Topological Structures 2. The Interpolation Theorem 3. Preservation and Definability 4. The Logic ℒtω1ω 5. Some Applications 6. Other Structures Chapter XVI. Borel Structures and Measure and Category Logics 1. Borel Model Theory 2. Axiomatizability and Consequences for Category and Measure Logics 3. Completeness Theorems Part F. Advanced Topics in Abstract Model Theory Chapter XVII. Set-Theoretic Definability of Logics 1. Model-Theoretic Definability Criteria 2. Set-Theoretic Definability Criteria 3. Characterizations of Abstract Logics 4. Other Topics Chapter XVIII. Compactness, Embeddings and Definability 1. Compact Logics 2. The Dependence Number 3. ℒ-Extensions and Amalgamation 4. Definability Chapter XIX. Abstract Equivalence Relations 1. Logics with the Robinson Property 2. Abstract Model Theory for Enriched Structures 3. Duality Between Logics and Equivalence Relations 4. Duality Between Embedding and Equivalence Relations 5. Sequences of Finite Partitions, Global and Local Back-and-Forth Games Chapter XX. Abstract Embedding Relations 1. The Axiomatic Framework 2. Amalgamation 3. ω-Presentable Classes Bibliography