Classes of barren extensions

TL;DR

This paper explores how various classes of non-Ramsey ultrafilters and certain Boolean algebras, when used as forcing notions, produce barren extensions that preserve many properties of the original model, extending previous results beyond Ramsey ultrafilters.

Contribution

It demonstrates that several classes of non-Ramsey ultrafilters and specific Boolean algebras also generate barren extensions, broadening the scope of models with preserved properties.

Findings

Non-Ramsey ultrafilters produce barren extensions similar to Ramsey ultrafilters.

Various classes of ultrafilters, including p-points and rapid ultrafilters, generate models with preserved properties.

Boolean algebras like in^{\u03b1} also produce barren extensions.

Abstract

Henle, Mathias, and Woodin proved that, provided that $\omega\rightarrow(\omega)^{\omega}$ holds in a model $M$ of ZF, then forcing with $([\omega]^{\omega},\subseteq^*)$ over $M$ adds no new sets of ordinals, thus earning the name a "barren" extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model $M[\mathcal{U}]$, where $\mathcal{U}$ is a Ramsey ultrafilter, with many properties of the original model $M$. This begged the question of how important the Ramseyness of $\mathcal{U}$ is for these results. In this paper, we show that several classes of $\sigma$-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken-Taylor ultrafilters, a class of rapid p-points of Laflamme, $k$-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares and Trujillo. Furthermore, the class of Boolean algebras $\mathcal{P}(\omega^{\alpha})/\mathrm{Fin}^{\otimes \alpha}$, $2\le \alpha<\omega_1$, forcing non-p-points also produce barren extensions.