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The category of complete boolean algebras with complete homomorphisms, and the basic theory of iterated forcing
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Academic year 2019/2020
- Teacher
- Prof. Matteo Viale (Lecturer)
- Type
- Basic
- Delivery
- Formal authority
- Language
- Italian
- Attendance
- Obligatory
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Sommario del corso
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Program
The course starts detailing the basic algebraic properties of complete boolean algebras and complete homomorphisms, it then links these properties to the method of forcing devised by Cohen to produce independence results in set theory.
We will briefly recall the basic results about boolean valued models for set theory; then we will develop the basic theory of iterated forcing attaching in a functorial way to any directed system of complete homomorphisms of complete boolean algebras a corresponding system of forcing extensions and embeddings between them. We will outline how direct and inverse limits in the category of complete boolean algebras are reflected by this functorial relation in corresponding limits in the category of forcing extension with embeddings.
Time permitting, and depending on the interests of the students, we will explore some important set theoretic applications of these techniques: one possibility is to present Woodin’s generic absoluteness results for second order arithmetic, another is to present the classical results of Shelah on iteration of proper and semiproper forcings.
The students attending the course should have a basic knowledge of the forcing method (at the level one gains attending the set theory course of our master program) and an acquaintance with the basic results on boolean algebras such as: Stone duality, representation of complete boolean algebras as algebras of regular open sets, existence and uniqueness of the boolean completion of a partial order, etc.
The main bibliographic reference is a set of notes extracted by a draft of a book on these topics I’m preparing, other bibliographic references will be given in due course during the lectures.Suggested readings and bibliography
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