Théorie des ensembles : Outils classiques

Syllabus

Set theory means many things to many people, but most will agree that it is a subject of mathematics that has two main roles: a foundational role in the way it gives axioms to most (although not all) aspects of modern mathematics, and a mathematical role in the way that it provides a firm theory of infinity. In its most common axiomatisation, Zermelo-Fraenkel with Choice (ZFC), set theory has been able to join these two roles to the point that each describes the strength and the limits of the other. These are best understood through the study of inner and outer models of set theory, notably the constructible universe L and the method of forcing.

This course will open the first pages of the advanced set theory, assuming that the student already knows basic axiomatic set theory. It will describeforcing, go through the classical proof that the Continuum Hypothesis is not a consequence of the axioms of ZFC and then continue to give some more classical forcing notions, for example the Lévy Collapse. This will naturally lead to the study of iterated forcing and Martin’s Axiom, some applications and limitations.

Contents

• Class Models of ZFC and inner models, notably L
• Forcing Method and the consistency of the failure of CH
• Some other classical forcing models
• Martin Axiom MA
• Applications and limitations of MA

Bibliography

• P. Dehornoy, Théorie des ensembles, Calvage et Mounet 2017
• M. Džamonja, Fast Track to Forcing, Cambridge University Press 2020
• T. Jech, Set Theory, 3rd Millenial Edition, Springer-Verlag, 2003
• K. Kunen, Set Theory with Introduction to Independence Proofs, North-Holland 1980

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