## Advanced Topics in Mathematical Logic: Forcing (V5A6)

### Lecturer

- Dr. Philipp Lücke
- Dr. Philipp Schlicht

### Time and place

- Lecture: Tuesday 14.15-16.00 in room 1.007 and Wednesday 10.15-12.00 in room 0.007.

### Contents

The method of forcing was introduced by Paul Cohen in his proof of the independence of the Continuum Hypothesis and the Axiom of Choice. Forcing proved to be a remarkably general technique for producing a large number of models and consistency results. In this lecture, we present several important forcing constructions, in particular

- The consistency of PFA.
- Solovay's model of ZF+DC+"all sets of reals are Lebesgue-measurable" (Introduction, Intermediate models of forcing extensions, The Levy Collapse, Borel Codes, The proof of Solovay's theorem).
- Singularizing cardinals by forcing: Prikry forcing, Namba forcing and variations.
- Indestructibility of large cardinals.
- Failures of the Singular Cardinal Hypothesis from a supercompact cardinal.
- Precipitous and saturated ideals.
- The nonstationary ideal.
- Adding large cardinals by forcing (Generic square sequences, Threading square sequences by forcing).
- Stationary reflection (Introduction, Club shooting, Proof of the Harrington-Shelah reflection result, Simultaneous reflection).
- The tree property at successor cardinals.

The exams are on Friday, July 11 and Friday, August 01. The second exam is on Thursday, September 11.