Perfect Tree Forcings for Singular Cardinals

We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. $\mathbb{P}$ satisfies a Sacks-type property, implying that $\mathbb{B}$ is $(ω,\infty,<κ)$-distributive. The $(\mathfrak{h},2)$-d.l. and the $(\mathfrak{d},\infty,<κ)$-d.l. fail in $\mathbb{B}$. $\mathcal{P}(ω)/\mbox{Fin}$ completely embeds into $\mathbb{B}$. Also, $\mathbb{B}$ collapses $κ^ω$ to $\mathfrak{h}$. We further prove that if $κ$ is a limit of countably many measurable cardinals, then $\mathbb{B}$ adds a minimal degree of constructibility for new $ω$-sequences. Some of these results generalize to cardinals $κ$ with uncountable cofinality.

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