# Large cardinals need not be large in HOD

## Joel David Hamkins

### The City University of New York

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.

Professor Hamkins (Ph.D. 1994 UC Berkeley) conducts research in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite. He has been particularly interested in the interaction of forcing and large cardinals, two central themes of contemporary set-theoretic research. He has worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess. His work on the automorphism tower problem lies at the intersection of group theory and set theory. Recently, he has been preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as in his work on the modal logic of forcing and set-theoretic geology.