This will be a talk for the CUNY Set Theory Seminar, January 31, 2014, 10:00 am.

**Abstract.** I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I will begin with the example of a measurable cardinal that is not measurable in HOD. After this, I will describe how to force a more extreme divergence. For example, among other possibilities, it is relatively consistent that there is a supercompact cardinal that is not weakly compact in HOD. This is very recent joint work with Cheng Yong.

Hi, Joel. How did the talk go?

It went very well, Everett, thank you for asking. I spent most of the time on the case of a measurable cardinal that is not measurable in HOD, but also had time to indicate how to make a supercompact cardinal that is not weakly compact in HOD, and indeed how to do this with a proper class of supercompact cardinals. The paper is still in-progress, but I will surely post it here on my web page when it is ready.

Is your proofs based on Kunen trick that at wekly compact cardinals, $Add(\kappa, 1)$ can be written as a two step iteration using higher Souslin trees

Yes, for the case where the cardinal is not weakly compact in HOD, this is part of it.

I have heard of this trick, but don’t recall having seen the proof anywhere. Would either of you point me to an article/book/online resource where I can get some more details, please?

It first appeared in “Kunen, Kenneth, Saturated ideals. J. Symbolic Logic 43 (1978), no. 1, 65–76.” You can also see Cummings paper in Handbook of set theory.

This method is also used in my paper http://jdh.hamkins.org/least-weakly-compact/, and the account in Victoria Gitman and P. D. Welch. Ramsey-like cardinals II. J. Symbolic

Logic, 76(2):541–560, 2011 (http://boolesrings.org/victoriagitman/2009/07/31/ramsey-like-cardinals-ii/) is particularly clear.

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