Topic Archive: Forcing and Large Cardinals
This will be the dissertation defense of the speaker. There will be a one-hour presentation, followed by questions posed by the dissertation committee, and afterwards including some questions posed by the general audience. The dissertation committee consists of Joel David Hamkins (supervisor), Gunter Fuchs, Arthur Apter, Roman Kossak and Philipp Rothmaler.
I will present the result of Laver demonstrating that the existence of a supercompact cardinal implies the existence of a Laver function, and using this to construct a model with a supercompact cardinal k which remains supercompact in any k-directed-closed forcing extension.
I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I shall begin with the example of a measurable cardinal that is not measurable in HOD, and then afterward describe how to force more extreme examples, such as a model with a supercompact cardinal, which is not weakly compact in HOD. This is very recent joint work with Cheng Yong.