After successful completion of the course, students are able to...
explain and judiciously apply the following concepts: properties of generic reals, composition of forcing notions, direct and projective limit of forcing iterations, preservation theorems (in particular: chain conditions, properness, w^w-bounding, axiom A)
We first review the main concepts and theorems from the forcing method: dense sets, maximal antichains, names, generic filters, the forcing theorem. Then we consider several classical examples of forcing notions such as Cohen, random, Sacks, Mathias, as well as finite products and compositions of forcing notions.
The main topics will be transfinite iterations of forcing notions, with finite and countable supports (ccc or proper iterations, respectively). Applications: Suslin's conjecture, Martin's Axiom, the Borel conjecture.
If some time is left, we will also consider the Levy collapse and Solovay's models.
