Laurea magistrale in Matematica

Familiarity with first order logic (compactness theorem, completeness theorem, Lowenheim-Skolem theorems); familiarity with the basics of set theory (ZFC as its first order axiomatizion, recursion theorem, basic results on cardinals and ordinals, basic results on the structure of the universe, its cumulative hierarchy, Mostowski collpasing theorem).
See the moodle page of the course for details.

The goal of the course is to show how set theory allows us to obtain efficient techniques to prove the undecidability of certain problems arising in various fields of mathematics, including the most abstract parts of analysis and topology

The student should be able to master the various aspects of the arguments presented among which: the basic features of the universe of sets, the theory of boolean valued models, the constructible sets, the forcing method, the undecidability of the continuum problem

1. Boolean algebras, Stone duality, boolean completion of partial orders.

2. Basic combinatorics for partial orders.

3. Boolean valued models and boolean valued semantics, full models and Los theorem.

4. Absoluteness and transitive models of fragments of ZFC.

5. Forcing via boolean valued models.

6. Indipendence of the continuum hypothesis from the standard axioms fo set theory.

See the moodle webpage for details.

The moodle page of the course will contain a rich set of material very useful also for those not able to attend in person. The course consists of 48 hours lectures (2 lectures of 2 hours per week for approximately 12 weeks).

 

Oral exam.

See the moodle page for details.

The votes range from 0 to 30 (cum laude)

Set Theory: An Introduction to Independence Proofs - Ken Kunen.

Notes of the teacher available on the moodle webpage.