Classes of 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.