Henle, Mathias, and Woodin proved in [21] 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.

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贫瘠的扩展课程

Henle、Mathias 和 Woodin 在 [21] 中证明，只要${\omega }{\rightarrow }({\omega })^{{\omega }}$持有模型米ZF 的，然后用$([{\omega }]^{{\omega }},{\subseteq }^*)$超过米没有添加新的序数集，因此获得了“贫瘠”的扩展名。此外，在一个额外的假设下，他们证明了这个通用扩展保留了所有强分区基数。这种强迫因此产生了一个模型$M[\mathcal {U}]$， 在哪里$\数学{U}$是 Ramsey 超滤器，具有原始模型的许多特性米. 这就引出了一个问题，即 Ramseyness 的重要性$\数学{U}$是为了这些结果。在本文中，我们展示了几类$\西格玛$- 产生非 Ramsey 超滤器的闭合力具有相同的特性。此类超滤器包括 Milliken-Taylor 超滤器，一类快速 p 点的 Laflamme，ķ-鲍姆加特纳和泰勒的箭头 p 点，以及由 Dobrinen、Mijares 和 Trujillo 构建的一类超滤器的扩展。此外，布尔代数类$\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$,$2\le {\alpha }<{\omega }_1$，强制非 p 点也会产生贫瘠的延伸。