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

本文探讨了与伍丁 HOD 猜想相关的几个主题。我们将伍丁 HOD 二分定理的大基数假设从可扩展基数改进为强紧基数。我们证明，假设存在一个强紧基数并且 HOD 假设成立，则不存在从 HOD 到 HOD 的基本嵌入，从而解决了 Woodin 问题。我们证明 HOD 假设相当于累积层次结构的级别的基本嵌入的唯一性属性。我们证明 HOD 假设成立当且仅当第一个强紧基数之上的每个正则基数都带有可定义的序数$\omega$-琼森代数。我们证明，如果 HOD 假设成立并且 HOD 满足超幂公理，则每个超紧基数在 HOD 中都是超紧的。