## Graduate Seminar on Set Theory - Large Cardinals and Forcing (S4A4)

### Organizers

- Prof. Dr. Peter Koepke
- Dr. Bernhard Irrgang

### Time and place

Tuesday 10-12, EA60 007.

### Contents

In many cases the consistency of a combinatorial statement cannot be proved from the consistency of ZFC alone. Often one has to make stronger assumptions like the existence of a model of ZFC in which there exists an inaccessible cardinal. For example, given such a model there exists a generic extension of it in which there are no Kurepa trees. Conversely, if there exist no Kurepa trees, then omega_2 is inaccessible in Gödel's constructible universe L. This shows that the consistency of an inaccessible cardinal is necessary.

Large cardinals are generalisations of inaccessible cardinals. In the seminar we will look at various situations where one has to assume the existence of large cardinals in the ground model to prove the consistency of a combinatorial statement by forcing.

### Talks

- Jeff Serbus:
**Inaccessibles and Kurepa trees I**, Jech 551 and Exercise 27.5 **Inaccessibles and Kurepa trees II**, Jech 553- Cecilia Bohler:
**Weakly compact cardinals and Aronszajn trees**, Jech 569 **Mahlo cardinals and square**, Exercise 27.2- Katharina Schinagl:
**Measurable cardinals** **The Levy-Solovay theorem**, Jech 389 - 392- Philipp Bongartz.
**Kunen-Paris forcing**, Jech 392 - 394 - Anne Fernengel:
**Prikry forcing**, Jech 400 - 403

### References

T. Jech: Set Theory, Third Millenium Edition, Springer 2002