I'm wondering if there are any known result for the maximum large cardinal strength which can be preserved by Radin forcing? For instance, with any large cardinal hypothesis in the ground model, can one show that in $$V[G]$$ which is a generic for a Radin forcing of some length $$\kappa$$ is $$V_{\kappa + 2}$$-strong?
If you start with a strong (or a supercompact cardinal) and if you force with Radin forcing $$\mathbb{R}_u$$, for some suitable $$u$$, then you can preserve the full strength (or supercompactness) of $$\kappa$$.
• Thanks for the references. I'm sorry for the ambiguity of my question. What I actually want to know is that, when you have a supercompact $\kappa$, but you can only use the measure sequence on $V_\kappa$ (i.e. not the supercompact Radin forcing associate to $P_\kappa(\lambda)$), how much strength can you preserved, can you find a particular length to show that $\kappa$ is strong. – Jiachen Yuan Apr 10 '19 at 6:04
• For ordinary Radin forcing, you can preserve all supercompactness even, see Radin's paper (he shows you can preserve $\lambda$-supercompactness). My paper with Gitik also uses the ordinary Radin forcing and not its supercompact version. Similar facts about strong cardinals. – Mohammad Golshani Apr 10 '19 at 6:08