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1

Generally, "the full Solovay model" means that there was $\kappa$ which was inaccessible in $M$ and $G$ is a generic filter for $\operatorname{Coll}(\omega,<\kappa)$. Since this is a homogenous forcing, it doesn't change $\rm HOD$, so $\rm HOD$ of $M[G]$ is the same as in $M$. If $\kappa$ is inaccessible in $M$ then it is regular and strong limit. In ...

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Clearly every set in the ultrafilter is stationary, that is part of the definition of a filter to begin with; but on the other hand, if a set has non-empty intersection with all the members of an ultrafilter it has to be there as well, otherwise its complement is in the ultrafilter... Kanamori's definition is more general, because it works for filters, ...

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It's pretty straightforward to show that if $\kappa$ is a 2-strong cardinal$^*$, then $V_\kappa$ thinks that for all $\alpha$, there is an $\alpha$-measurable cardinal. In particular, we can show by induction that for $\alpha<\kappa$, $\kappa$ is $\alpha$-measurable. Let $\mathcal U$ be the normal ultrafilter on $\kappa$ defined by $X\in \mathcal U$ iff ...

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