|Weekly hours||4 h CM|
|Years||Master Logique Mathématique et Fondements de l'Informatique M2 Logos|
On 8 August 1900, at the Second International Congress of Mathematicians in Paris, David Hilbert set out a list of 23 mathematical problems which, in his opinion, should serve as a guide for future research in the new century. The first problem in this list, Cantor's continuum hypothesis, was solved in two stages: by Gödel (1938) who constructed an internal model of the generalized continuum hypothesis, and by Paul Cohen (1963), who invented a model construction for the negation of Cantor's hypothesis. This course will mainly cover the two model constructions of set theory introduced by Gödel and Cohen.
- Review of basic set theory: cardinals, ordinals, partial orders, Boolean algebras, etc.
- Trees and Ramsey theory.
- models of ZFC, reflection, relativizations, ordinal definable sets.
- Gödel's constructible universe L, the consistency of the Axiom of Choice and the Continuum Hypothesis.
- Notion of forcing and generic extensions, fundamental theorem of forcing.
- Applications of forcing: the "diamond" principle, Souslin and Kurepa trees, etc.
- Iterated forcing, Martin's axiom and applications.
- P. Dehornoy, Théorie des ensembles, Calvage et Mounet 2017
- T. Jech, Set Theory, 3rd Millenial Edition, Springer-Verlag, 2003
- K. Kunen, Set Theory with Introduction to Independence Proofs, North-Holland 1980
- N. Weaver, Forcing for mathematiicians (World Scientific 2014)