Around the HOD dichotomy (ASPEROD_U26EMP)

HOD, the class of sets which are hereditarily definable from ordinals, is a particularly interesting subuniverse of ZFC one can always build in ZF. Woodin’s HOD dichotomy asserts that, in the presence of suitable large cardinals, HOD is either close to the real universe in some well-defined sense, or else it is much smaller than the real universe. This dichotomy, and the accompanying HOD Conjecture, is currently an active area of research, and figures prominently in contemporary foundational debates in set theory, e.g. around Woodin’s ultimate-L programme.

There are still many open questions, for example regarding the prospects for falsifying the HOD Conjecture, or concerning possible patterns of $\omega$-strongly measurable cardinals in HOD. Typical objects one encounters and uses in this area of enquiry are choiceless large cardinals, and forcing over ZFC models with large cardinals. This project is offered on a self-funding basis.

It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found here. A bench fee is also payable in addition to the tuition fee to cover specialist equipment or laboratory costs required for the research.

Applicants should contact the primary supervisor for further information about the fee associated with the project.