Strongly Compact Cardinals and Ordinal Definability

This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of dichotomy theorem from an extendible a strongly compact cardinal. show that assuming there is and holds, no elementary embedding HOD, settling question Woodin. equivalent uniqueness property embeddings levels cumulative hierarchy. prove holds if only every regular above first carries ordinal definable ([Formula: see text])-Jónsson algebra. satisfies Ultrapower Axiom, then supercompact in HOD.