Ordinal definable subsets of singular cardinals

2 Responses to Ordinal definable subsets of singular cardinals

1. Mohammad says:

   April 17, 2016 at 12:34 am

   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 $p^{\prime }\le p,q^{\prime }\le q$ and an isomorphism from $P/p^{\prime }$ onto $P/q^{\prime }$). 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 $Col(\kappa ,<\alpha )\ast P_{U_{\alpha }}$ 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 $Col(\kappa ,<\alpha )\ast P_{U_{\alpha }}$ , then it seems we are done. This seems reasonable as the whole forcing is $\lambda$-c.c.

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2. saf says:

   July 18, 2017 at 3:12 pm

   Submitted to Israel Journal of Mathematics, April 2016.
   Accepted July 2017.

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