Set theory, independence proofs and the continuum hypothesis

The course will provide an introduction to set theory and independence proofs, with the goal of ultimately proving that the famous Continuum Hypothesis is independent of Zermelo Fraenkel axioms of set theory.

The course will be made as accessible as possible (and will review basic notions from logic and model theory), but at least some previous knowledge of first-order logic will be useful.

Note: Course is given in English.

Relationship of logic, set theory and mathematics: review of basic notions of logic (proof system, completeness theorem, Lowenheim-Skolem theorem, etc), discussion of mathematics vs metamathematics (2-3 lectures)

Zermelo Fraenkel set theory: axioms, well-ordered sets, ordinals, transfinite induction and recursion (3-4 lectures)

Independence proofs via inner models: independence of the axiom of regularity, constructible universe, absoluteness, consistency of the axiom of choice and continuum hypothesis (4-5 lectures)

Forcing: models of set theory (standard models, minimal model), forcing as a technique, independence of the axiom of choice and the continuum hypothesis (4-5 lectures)

Book “Set Theory and the Continuum Hypothesis” by Paul Cohen

Book “Set Theory: An Introduction to Independence Proofs” by Kenneth Kunen

Lecture notes provided by the lecturer

The student will have understanding of basic notions of set theory, will be able to prove independence results and have a basic working knowledge of the forcing method

Oral exam

The course can provide credit for doctoral students as a "methodological course".

In that case, there is an additional requirement for passing the course: The student should correctly solve an assignment given by the lecturer, or study and present a result assigned by the lecturer.