A Crèche Course in Model Theory. Lecture notes for an introductory (under)graduate couse in model theory
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README.md

A Crèche Course in Model Theory

Lecture notes for an introductory (under)graduate course

Printable output in file creche.pdf

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
    • \L o'{s}'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
  • Fraïssé limits
    • Rich models.
    • 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
    • Immaginaries: the true story
    • Uniform elimination of imaginaries
    • Notes and references
  • Invariant sets
    • Invariant sets and types
    • Invariance from the dual perspective
    • Heirs and coheirs
    • Morley sequences and indiscernibles
  • Ramsey theory
    • Ramsey theorem from coheir sequences
    • Ehrenfeucht-Mostowski construction of indiscernibles
    • Indempotent 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
    • Vapknik-Chervonenkis dimension
    • Honest definitions
    • Stable theories
    • Stability and the number of types
    • Notes and references
  • Keisler measures
    • T.b.c.