Perfect Tree Forcings for Singular Cardinals

TL;DR

This paper studies perfect tree forcings associated with singular cardinals, revealing their distributivity properties, embeddings, and effects on cardinal characteristics, thus advancing understanding of forcing at singular cardinals.

Contribution

It provides new results on the distributivity, embeddings, and collapsing properties of perfect tree forcings for singular cardinals, including generalizations to uncountable cofinality.

Findings

The Boolean completion is $( ext{omega}, ext{mu})$-distributive for all $ ext{mu}< ext{kappa}$.

The Boolean algebra satisfies a Sacks-type property, ensuring $( ext{omega}, ext{infinity},< ext{kappa})$-distributivity.

The algebra collapses $ ext{kappa}^ ext{omega}$ to $ ext{h}$ and embeds $ ext{P}( ext{omega})/ ext{Fin}$.

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\kappa$, implying failure of the $(\omega,\kappa)$-d.l. These hitherto unpublished results are included, setting the stage for the following recent results.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.