CLASSES OF BARREN EXTENSIONS

Henle, Mathias, andWoodin proved in [21] that, provided that omega ->(omega)(omega) holds in amodel M of ZF, then forcing with ([omega](omega), subset of*) 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 [U], where U is a Ramsey ultrafilter, with many properties of the original model M. This begged the question of how important the Ramseyness of 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 P(omega(alpha))/Fin(circle times alpha), 2 <= alpha < omega(1), forcing non-p-points also produce barren extensions.