Teaching material

Elementi di logica matematica
Author: Alessandro Andretta

Comments: 488 pages, in Italian (pdf).

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Notes for the course "Logica" (degree in mathematics, third year).

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A crèche course in model theory
Author: Domenico Zambella

Comments: Lecture notes for an introductory (under)graduate course (pdf).

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Table of contents

  • 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
    • Ultraproducts
  • 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
    • Notes and references
  • 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
  • Definability and automorphisms
    • 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
    • Notes and references
  • Invariant sets
    • Invariant sets and types
    • Invariance from the dual perspective
    • Morley sequences and indiscernibles
    • From coheirs to Ramsey to indiscernibles
  • 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
    • Vaknik-Chevronenkis dimension
    • Honest definitions
    • Stable theories
    • Stability and the number of types
    • Notes and references
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Other notes

A boolean algebraic approach to semiproper iterations
Authors: Giorgio Audrito, Silvia Steila and Matteo Viale

Comments: 60 pages, in English (pdf).

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The paper gives a self contained development of iterated forcing by means of directed systems of boolean algebras and proves the main results on semiproper iterations expanding and developing on the work of Donder and Fuchs. This material expands on a Ph.D course I gave in summer and fall 2013 on this topic.

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Slides

Logica matematica
Author: Luca Motto Ros

Comments: 57 slides, in Italian (pdf).

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Slides of the first two (out of four) introductory seminars on Mathematical Logic held at the University of Milano Bicocca (May 5-6 2016).

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Calcoli logici
Author: Riccardo Camerlo

Comments: in Italian (pdf).

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Slides of the third (out of four) introductory seminars on Mathematical Logic held at the University of Milano Bicocca (May 12 2016).

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Teoria degli insiemi
Author: Riccardo Camerlo

Comments: in Italian (pdf).

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Slides of the fourth (out of four) introductory seminars on Mathematical Logic held at the University of Milano Bicocca (May 13 2016).

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