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?
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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 $\kappa$.
To preserve partial strength, (weak) repeat points are sufficient (see Radin's paper Adding closed cofinal sequences to large cardinals).
To get the full strength, see for example the proof of Theorem 2.14 of my paper with Gitik Adding a lot of Cohen reals by adding a few I.
answered Apr 10, 2019 at 5:52
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$\begingroup$ 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. $\endgroup$ Commented Apr 10, 2019 at 6:04
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$\begingroup$ 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. $\endgroup$ Commented Apr 10, 2019 at 6:08