Model Theory

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Course Description

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Notes

Basics

• First Order Logic
  • Syntax of First Order Logic
  • Semantics of First Order Logic
• Theories
• Examples of Complete and Incomplete Theories
  • Example of Atomless Boolean Algebra
• Homomorphism of Models
• Product of Models
• Reduced Product
  • Filters

Compactness

• Łoś’s Theorem
• Finiteness Theorem
  • Upward Löwenheim–Skolem Theorem
    • Downward Löwenheim–Skolem Theorem
  • Class of Periodic Groups is not Axiomatizable
  • Class of Division Rings with Finite Order is not Axiomatizable
  • Stone’s Representation Theorem
  • Compactness Theorem
• Tarski-Vaught test
• Elementary Equivalence of Models
• Theory of torsion-free divisible Abelian Groups is kappa categorical for uncountable kappa
• ACFp is kappa categorical for uncountable kappa
• Vaught Test
• Atomic Diagram of a structure
• A Theory is universally axiomatiazble iff its closed under substructure
• Marczewski-Szpilrajn Theorem (WO)
• Marczewski-Szpilrajn Theorem (Compactness)

Quantifier Elimintation

• There is a unique countable dense linear order upto isomorphism
• Prime Models
  • Existence of Algebraically Prime Models
  • Theories that admit Algebraically Prime Models
• Simply Closed sub-models
• A Theory T admits QE for phi if any 2 models of T agree on phi on intersection

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MOCs

• Logic
• Category Theory
• Topology
• Finite Model Theory

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Practicle Information

Profs : Suresh & Kummini Timings : Monday and Wednesday 10:30-11:45 Loc : LH 3

Books recommended:

• David Marker: An introduction
• Chang and Keisler: Model Theory
• Bruno Poizat : A course in model theory (idiosyncratic but pretty)
• Tent and Ziegler : A course in model theory

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References