Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova.
Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such that $\text{HOD}_x$ contains the power set of $\kappa$.
In this paper, we develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal $\kappa$ of countable cofinality, $\kappa^+$ is supercompact in $\text{HOD}_x$ for all $x \subseteq\kappa$.
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Citation information:
J. Cummings, S.-D. Friedman, M. Magidor, A. Rinot and D. Sinapova, Ordinal definable subsets of singular cardinals, Isr. J. Math., 226(2), 781-804, 2018.
A nice result. As it is stated at the end of your paper, the “supercompact extender based Prikry forcing ” of Merimovich works as well. The point is that the forcing is cone homogeneous ($P$ is cone homegeneous if given p, q, there are $p’ \leq p, q’ \leq q$ and an isomorphism from $P/p’$ onto $P/q’$). Also any subset of $\kappa$ is forced by some subforcing of small size.
I am wondering if the following works or not: Let $\kappa < \lambda,$ with $\kappa$
supercompact Laver indestructible and $\lambda$ measurable. Force with $Col(\kappa, < \lambda)$ and with Prikry forcing $P_U$ over it, for some normal measure $U$ on $\kappa.$ Let $\dot{U}$ be a $Col(\kappa, < \lambda)$-name for $U$. Note that there are many $\alpha < \lambda$ such that $Col(\kappa, < \alpha)*P_{U_\alpha}$ is a subforcing of the final forcing, where $U_\alpha$ is $U \cap V[Col(\kappa, < \alpha)].$
If someone can show that any subset of $kappa$ is in fact in some extension by $Col(\kappa, < \alpha)*P_{U_\alpha}$ , then it seems we are done. This seems reasonable as the whole forcing is $\lambda$-c.c.
Submitted to Israel Journal of Mathematics, April 2016.
Accepted July 2017.