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Abstract

We answer a question of Krueger by obtaining – from countably many Mahlo cardinals – a model where there is a disjoint stationary sequence on $\aleph_{n+2}$ for every $n\in \omega $. In that same model, the notions of being internally stationary and internally club are distinct on a stationary subset of $[H(\Theta )]^{\aleph_{n+1}}$ for every $n\in \omega $ and $\Theta \geq \aleph_{n+2}$, answering another of Krueger’s questions. This is obtained by employing a product of variants of Mitchell forcing which uses finite support for the Cohen reals and full support for the countably many collapses.