Perfect Tree Forcings for Singular Cardinals

Abstract

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 $\langle \kappa_n: n< \omega \rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<\omega}\kappa_n$, and proved that for $\kappa=\sup_{n<\omega}\kappa_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(\omega,\mu)$-distributive for all $\mu<\kappa$ but $(\omega,\kappa,\delta)$-distributivity fails for all $\delta<\kappa$, implying failure of the $(\omega,\kappa)$-d.l. These hitherto unpublished results are included, setting the stage for the following recent results.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 $(\omega,\infty,<\kappa)$-distributive. The $(\mathfrak{h},2)$-d.l. and the $(\mathfrak{d},\infty,<\kappa)$-d.l. fail in $\mathbb{B}$. $\mathcal{P}(\omega)/\mbox{Fin}$ completely embeds into $\mathbb{B}$. Also, $\mathbb{B}$ collapses $\kappa^\omega$ to $\mathfrak{h}$. We further prove that if $\kappa$ is a limit of countably many measurable cardinals, then $\mathbb{B}$ adds a minimal degree of constructibility for new $\omega$-sequences. Some of these results generalize to cardinals $\kappa$ with uncountable cofinality.