Lecture notes for an introductory (under)graduate course
Printable output in file PDF/creche.pdf
- Preface
- Preliminaries and notation
- Structures
- Tuples
- Terms
- Substructures
- Formulas
- Yet more notation
- Theories and elementarity
- Logical consequences
- Elementary equivalence
- A nonstandard example
- Embeddings and isomorphisms
- Quotient structures
- Completeness
- The Tarski-Vaught test
- Downward Löwenheim-Skolem
- Elementary chains
- Ultraproducts
- Filters and ultrafilters
- Direct products
- Łoś's Theorem
- Compactness
- Compactness via syntax
- Compactness via ultraproducts
- Compactness for types
- Finite axiomatizability
- Types and morphisms
- Semilattices and filters
- Distributive lattices and prime filters
- Types as filters
- Morphisms
- Some relational structures
- Dense linear orders
- Random graphs
- Notes and references
- Rich models
- Models and morphisms
- The theory of rich models and quantifier elimination
- Weaker notions of universality and homogeneity
- The amalgamation property
- Some algebraic structures
- Abelian groups
- Torsion-free abelian groups
- Divisible abelian groups
- Commutative rings
- Integral domains
- Algebraically closed fields
- Hilbert's Nullstellensatz
- Saturation and homogeneity
- Saturated structures
- Homogeneous structures
- The monster model
- Preservation theorems
- Lyndon-Robinson Lemma
- Quantifier elimination by back-and-forth
- Model-completeness
- Geometry and dimension
- Algebraic and definable elements
- Strongly minimal theories
- Independence and dimension
- Countable models
- The omitting types theorem
- Prime and atomic models
- Countable categoricity
- Small theories
- A toy version of a theorem of Zil'ber
- Notes and references
- Imaginaries
- Many-sorted structures
- The eq-expansion
- The eq-definable closure
- The eq-algebraic closure
- Elimination of imaginaries
- Examples
- Imaginaries: the true story
- Invariant sets
- Invariant sets and types
- Invariance from a dual perspective
- Heirs and coheirs
- Morley sequences and indiscernibles
- Ramsey theory
- Ramsey's theorem from coheir sequences
- The Ehrenfeucht-Mostowski theorem
- Idempotent orbits in semigroups
- Hindman theorem
- The Hales-Jewett Theorem
- Notes and references
- Lascar invariant sets
- Expansions
- Lascar strong types
- Coheirs over sets
- The Lascar graph and Newelski's theorem
- Kim-Pillay types
- Notes and references
- Externally definable sets
- Approximable sets
- Ladders and definability
- Stable theories
- Stability and the number of types
- Vapnik-Chervonenkis theory
- Vapnik-Chervonenkis dimension
- Honest definitions \contentsline {chapter}{References**