Topics in Mathematics Logic

The course is also offered to exchange programme students.

This class will cover Model Theory, Set Theory, and Computability Theory. Every time the class is offered it will cover one of the three subjects starting with Model Theory→ Set Theory→ Computability Theory→ Model Theory→ Set Theory→ ... For 2021 the subject to be covered is Model Theory. Below are the descriptions of all three subjects. Model Theory This is a Master's level introduction to Model Theory with applications from Algebra. The topics include:\ • Definable sets • Complete Theories, Algebraically Closed Fields • Upwards and Downwards Lowenheim-Skolem Theorem • Dense linear orders and back-and-forth arguments • Quantifier Elimination • Types, Omitting Types Theorem • Prime and Atomic Models • Saturated and Homogeneous Models • Vaught's Two-Cardinal Theorem • ω-Stable Theories and Morley's Theorem Bibliography: D. Marker, Model Theory: An Introduction We will cover parts of the first 6 chapters Prerequisites: Familiarity with Mathematical Logic and Set Theory (undergraduate level courses) Set Theory This class is an introduction to forcing and independence proofs. We will prove the independence of the Continuum Hypothesis (plus more). Topics include: • The axioms of ZFC • Boolean Algebras • Filters, Ultrafilters, and Generic Filters. • Boolean-valued Models • Generic Extensions • Forcing Theorem and Generic Extension Theorem • Countable Chain Condition and Preserving Cardinals • Independence of the Continuum Hypothesis and Independence of the Axiom of Choice Bibliography: T. Jech, Set Theory In this class we will focus on chapters 14 and 15. Prerequisites: Set Theory, Mathematical Logic (undergraduate courses) Computability Theory Computability Theory is a branch of Mathematics and, in particular, Mathematical Logic. The central problem in Computability Theory is the mathematical formulation of the notion of "algorithm" and "computable function". The notions of algorithm and computable function are interesting both for Mathematics, e.g. Hilbert's 10th problem, but also for Computer Science, e.g. the Halting problem. Topics include: • Primitive Recursive Functions and General Recursive Functions. • Recursion and Computation, Recursive Sets • Church- Turing Thesis • Turing machines; Turing computable functions • Semi-recursive functions and recursive enumerable sets • Kleene's normal form • Recursion Theorems • Arithmetical Hierarchy • Non-definability of Truth Bibliography: 1. Michael Sipser: Introduction to the Theory of Computation 3. Harry Lewis, Christos Papadimitriou: Elements of the Theory of Computation Prerequisites: Mathematical Logic (undergraduate course)

12-02-2021