Henle, Mathias, and Woodin proved that, provided that $omegarightarrow(omega)^{omega}$ 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 $mathcal{P}(omega^{alpha})/mathrm{Fin}^{otimes alpha}$, $2le alpha<omega_1$, forcing non-p-points also produce barren extensions.
تحميل البحث