A Crèche Course in Model Theory

Table of contents

• Preface
• Preliminaries and notation
  • Structures
  • Tuples
  • Terms
  • Substructures
  • Formulas
  • Yet more notation
• Theories and elementarity
  • Logical consequences
  • Elementary equivalence
  • 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
  • Upward Löwenheim-Skolem
  • 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
  • Rich models.
  • 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 definable closure in the eq-expansion
  • The algebraic closure in the eq-expansion
  • Elimination of imaginaries
  • Imaginaries: the true story
  • Uniform elimination of imaginaries
• 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
  • 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