Radek Honzik: Large cardinals and the continuum function

Week 1, Monday June 15, 11:45-12:10

By Easton's results, it is known that ZFC can prove very little about the continuum function. However, this result does not carry over directly to extensions of ZFC by large cardinals. This is due to the reflection properties of large cardinals, such as a measurable cardinal. We study the following problem using a cardinal-preserving forcing: given a universe of sets with large cardinals, which continuum functions are compatible with given large cardinals? This is a generalization of Easton's theorem to situations with large cardinals. As a bonus, we can extend these results naturally to include singular cardinals which will fail SCH, studying the compatibility of continuum functions and cardinals failing SCH.