6.2 Types and Realizations
Definition of complete and partial types, realization of types in structures, examples, consistency, and basic properties.
Tag
Model-Theory
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4.4 Isomorphism and Invariants
Isomorphisms of first order structures, structural invariants, and properties preserved by isomorphism.
4.2 Substructures and Embeddings
Substructures, generated substructures, homomorphisms, embeddings, and preservation of atomic formulas.
6.5 Classification Programs
Detailed overview of classification theory, dividing lines such as stability, simplicity, and NIP, and the structural analysis of first order theories.
6.4 Stability Theory Overview
Detailed introduction to stability theory, counting types, order property, definability of types, and structural consequences.
6.3 Saturated Models
Saturated models, realization of types, and their role in controlling definability and extensions.
6.1 Definable Sets and Functions
Definition of definable sets and functions in first order structures, with parameters, examples, closure properties, and proofs.
Chapter 6. Definability and Types
Definable sets, definable functions, types, realizations, saturated models, stability theory, and classification programs.
5.5 Limitations of First Order Logic
Expressive limitations of first order logic, including inexpressibility of finiteness and categoricity issues.
5.4 Nonstandard Models
Construction and properties of nonstandard models using compactness and Lowenheim Skolem.
5.3 Applications to Algebra
Applications of compactness and Lowenheim Skolem to algebraic structures and existence results.
5.2 Lowenheim Skolem Theorems
Downward and upward Lowenheim Skolem theorems and their consequences for model sizes in first order logic.
5.1 Compactness Theorem
Detailed development of the compactness theorem, its proof via completeness, and fundamental applications in model theory.
Chapter 5. Compactness and Completeness
Compactness, completeness, Lowenheim-Skolem theorems, nonstandard models, and limitations of first order logic.
4.5 Examples Across Algebra and Geometry
Examples of first order structures from algebra, order theory, graph theory, and geometry.
4.3 Elementary Equivalence
Elementary equivalence, theories of structures, and preservation of first order sentences.
4.1 Languages and Signatures
Formal languages, signatures, and symbols used to describe structures in first order logic.
Chapter 4. Structures and Models
Basic model theoretic notions including languages, signatures, substructures, embeddings, elementary equivalence, isomorphism, and examples.
Chapter 2. First-Order Logic
Extension of propositional logic with terms, predicates, quantifiers, structures, satisfaction, models, validity, and entailment.
08. General Algebraic Systems
Operations, identities, and structures as a unifying framework for all algebraic theories.