- Y. Cheng, S. Friedman, and J. D. Hamkins, “Large cardinals need not be large in HOD,” Annals of Pure and Applied Logic, vol. 166, iss. 11, pp. 1186-1198, 2015.

[Bibtex]`@ARTICLE{ChengFriedmanHamkins2015:LargeCardinalsNeedNotBeLargeInHOD, title = "Large cardinals need not be large in {HOD} ", journal = "Annals of Pure and Applied Logic ", volume = "166", number = "11", pages = "1186 - 1198", year = "2015", note = "", issn = "0168-0072", doi = "10.1016/j.apal.2015.07.004", eprint = {1407.6335}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://jdh.hamkins.org/large-cardinals-need-not-be-large-in-hod}, author = "Yong Cheng and Sy-David Friedman and Joel David Hamkins", keywords = "Large cardinals", keywords = "HOD", keywords = "Forcing", keywords = "Absoluteness ", abstract = "Abstract We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal κ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in V, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.", }`

**Abstract**. We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal $\kappa$ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.

In this article, we prove that large cardinals need not generally exhibit their large cardinal nature in HOD, the inner model of hereditarily ordinal-definable sets, and there can be a divergence in strength between the large cardinals of the ambient set-theoretic universe $V$ and those of HOD. Our general theme concerns the questions:

**Questions**.

1. To what extent must a large cardinal in $V$ exhibit its large cardinal properties in HOD?

2. To what extent does the existence of large cardinals in $V$ imply the existence of large cardinals in HOD?

For large cardinal concepts beyond the weakest notions, we prove, the answers are generally negative. In Theorem 4, for example, we construct a model with a supercompact cardinal that is not weakly compact in HOD, and Theorem 9 extends this to a proper class of supercompact cardinals, none of which is weakly compact in HOD, thereby providing some strongly negative instances of (1). The same model has a proper class of supercompact cardinals, but no supercompact cardinals in HOD, providing a negative instance of (2). The natural common strengthening of these situations would be a model with a proper class of supercompact cardinals, but no weakly compact cardinals in HOD. We were not able to arrange that situation, however, and furthermore it would be ruled out by Conjecture 13, an intriguing positive instance of (2) recently proposed by W. Hugh Woodin, namely, that if there is a supercompact cardinal, then there is a measurable cardinal in HOD. Many other natural possibilities, such as a proper class of measurable cardinals with no weakly compact cardinals in HOD, remain as open questions.