https://github.com/domenicozambella/creche

A Crèche Course in Model Theory. Lecture notes for an introductory (under)graduate couse in model theory
https://github.com/domenicozambella/creche

latex-document lecture-notes logic model-theory

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

• Host: GitHub
• URL: https://github.com/domenicozambella/creche
• Owner: domenicozambella
• License: gpl-3.0
• Created: 2017-03-20T11:16:00.000Z (over 8 years ago)
• Default Branch: master
• Last Pushed: 2024-10-26T17:48:35.000Z (about 1 year ago)
• Last Synced: 2024-10-26T22:24:16.464Z (about 1 year ago)
• Topics: latex-document, lecture-notes, logic, model-theory
• Language: TeX
• Homepage:
• Size: 58.9 MB
• Stars: 8
• Watchers: 3
• Forks: 1
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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# [A Crèche Course in Model Theory](../../raw/master/PDF/creche.pdf)

Lecture notes for an introductory (under)graduate course

Printable output in file [`PDF/creche.pdf`](../../raw/master/PDF/creche.pdf)

## Table of contents

*  **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**