Model Theory ‖ La Théorie des Modèles

Classical tools ‖ Outils classiques

2023–2024, Université Paris Cité
organization:
  • second semestre
  • Cours de Théorie des modèles : Outils classiques dans M2LMFI
  • 8 ECTS
  • 4 hours CM/week
  • outline lecture notes: preliminaries / part one
  • problem sheets: first batch
  • Partial exam: (date to be confirmed) ??th February/March 2025
books
  • David Marker: Model Theory: An introduction
  • Katrin Tent and Martin Ziegler: A course in Model Theory
  • Wilfred Hodges: A shorter Model Theory
course plan
  • Reminder (premier semestre):
    • Languages, structures, first-order theories
    • Ultraproducts and compactness
    • Embeddings, elementary embeddings, method of diagrams
    • Löwenheim–Skolem Theorems, elementary chains
    • Back and forth
    • Universal, existential, and inductive theories, preservation theorems
    • Model completeness and quantifier elimination
  • Part 1: Fundamental Theorems, types, and countable models
    • Week 1: Fundamental Theorems of Model Theory
      • The Compactness Theorem
      • The Löwenheim–Skolem Theorems
      • Beth’s Theorem
      • Separation Lemma
      • Ultraproducts and Łoś’s Theorem
      • Examples: infinite set, DLO, random graph, algebraically closed fields, vector spaces
    • Week 2: Countable models and omitting types
      • Types, partial versus complete, type spaces, and realising types
      • The Omitting Types Theorem
      • ($\kappa$-)categoricity, $\aleph_{0}$-categoricity
      • The Theorem of Engeler, Ryll-Nardzewski, and Svenonius
      • The Amalgamation Property, Ultrahomogeneity, and Fraïssé’s Theorem
    • Week 3: More on countable models
      • Back and forth again
      • Saturation and homogeneity
      • Prime and atomic models
      • Vaught’s “Not $2$” Theorem
      • $0$-stable theories
  • Part 2: Quantifier elimination and algebraic model theory
    • Week 4: Quantifier elimination and friends
      • Quantifier elimination
      • Model completeness
      • Criteria
    • Week 5: Theories of fields
      • algebraically closed fields, separably closed fields, real closed fields
      • Pseudofinite, PAC, large fields
    • Week 6: Groups, graphs, and friends
      • equivalence relations, graphs, and groups
      • Discrete linear orders
      • DTFAG
      • OAGs, $\mathbb{Z}$-groups
      • cross-cutting equivalence relations
  • Part 3: Uncountable categoricity
    • Week 7: Morley’s Theorem
      • Indiscernable sequences and sets
      • Totally transcendental theories
      • Morley’s Theorem
      • Baldwin–Lachlan Theorem
      • Morley rank and degree
      • Strongly minimal sets
    • Week 8: Strong minimality
      • Strongly minimal theories
      • $\omega$-stable theories
      • Stable theories
    • Week 9: The beginnings of Stability Theory -
  • Part 4: Towards Neostability Theory
Controls
  • partiel
  • examen terminal