6.2 Types and Realizations
Definition of complete and partial types, realization of types in structures, examples, consistency, and basic properties.
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Definition of complete and partial types, realization of types in structures, examples, consistency, and basic properties.
Isomorphisms of first order structures, structural invariants, and properties preserved by isomorphism.
Substructures, generated substructures, homomorphisms, embeddings, and preservation of atomic formulas.
Detailed overview of classification theory, dividing lines such as stability, simplicity, and NIP, and the structural analysis of first order theories.
Detailed introduction to stability theory, counting types, order property, definability of types, and structural consequences.
Saturated models, realization of types, and their role in controlling definability and extensions.
Definition of definable sets and functions in first order structures, with parameters, examples, closure properties, and proofs.
Definable sets, definable functions, types, realizations, saturated models, stability theory, and classification programs.
Expressive limitations of first order logic, including inexpressibility of finiteness and categoricity issues.
Construction and properties of nonstandard models using compactness and Lowenheim Skolem.
Applications of compactness and Lowenheim Skolem to algebraic structures and existence results.
Downward and upward Lowenheim Skolem theorems and their consequences for model sizes in first order logic.
Detailed development of the compactness theorem, its proof via completeness, and fundamental applications in model theory.
Compactness, completeness, Lowenheim-Skolem theorems, nonstandard models, and limitations of first order logic.
Examples of first order structures from algebra, order theory, graph theory, and geometry.
Elementary equivalence, theories of structures, and preservation of first order sentences.
Formal languages, signatures, and symbols used to describe structures in first order logic.
Basic model theoretic notions including languages, signatures, substructures, embeddings, elementary equivalence, isomorphism, and examples.
Extension of propositional logic with terms, predicates, quantifiers, structures, satisfaction, models, validity, and entailment.
Operations, identities, and structures as a unifying framework for all algebraic theories.