Beschreibung
Inhaltsangabe1. Introduction.- 2. Notation, elementary concepts: 2.1 Sets, classes, tuples, simple operations on sets.- 2.2 Binary relations, equivalence relations, functions.- 2.3 Orderings, ordinals, cardinals.- 2.4 Sequences.- 2.5 Direct product of families of sets.- 2.6 Relations of higher ranks.- 2.7 First-order logic (FOL).- 3. Basics from universal algebra: 3.1 Examples for algebras.- 3.2 Building new algebras from old ones (operations on algebras): 3.2.1 Subalgebra.- 3.2.2 Homomorphic image.- 3.2.3 Cartesian product, direct decomposition.- 3.2.4 Subdirect decomposition.- 3.2.5 Ultraproduct, reduced product.- 3.3 Variety characterization, quasi-variety characterization.- 3.4 Discriminator varieties.- 3.5 Boolean algebras.- 3.6 Boolean algebras with operators.- 4. General framework for studying logics: 4.1 Defining the framework.- 4.2 Concrete logics in the new framework.- 4.2.1 Distinguished logics.- 4.2.2 The finite variable fragment of FOL. Finite model theory.- 4.2.3 Summary.- 4.2.4 Solutions for some exercises of subsections 4.2.1 and 4.2.2.- 5. Bridge between the world of logics and the world of algebras: 5.1 Fine-tuning the framework.- 5.2 Algebraic characterizations of completeness and compactness properties.- 6. Generalizations.- 7. Further equivalence results (definability, interpolation, omitting types) 8. New kinds of logics 9. The guarded fragment and its finite variable hierarchy.- 10. Distinguished algebras of logic: 10.1 Cylindric algebras.- 10.2 Relation algebras, substitution cylindric algebras, polyadic algebras.- 10.3 Relativization. The non-square approach to ``taming logics''.- 10.4 Brief return to guarded fragment and related positive results.- 11. Applications of the algebraic results to logic (via our bridge).- 12. Connections with set theory and foundational thinking.- 13. Connections with abstract model theory.- 14. Extending the bridge to including logic of spacetime, connections with relativity.- 15. On the problems in the books Henkin-Monk-Tarski: Cylindric Algebras Parts I,II and Henkin-Monk-Tarski-Andreka-Nemeti.- Some further problems.
