Classes of Barren Extensions

Journal of Symbolic Logic 86 (1):178-209 (2021)
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Henle, Mathias, and Woodin proved in [21] that, provided that${\omega }{\rightarrow }({\omega })^{{\omega }}$holds in a modelMof ZF, then forcing with$([{\omega }]^{{\omega }},{\subseteq }^*)$overMadds 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 modelM. 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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References found in this work

Happy families.A. R. D. Mathias - 1977 - Annals of Mathematical Logic 12 (1):59.
Suitable extender models I.W. Hugh Woodin - 2010 - Journal of Mathematical Logic 10 (1):101-339.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
Forcing with filters and complete combinatorics.Claude Laflamme - 1989 - Annals of Pure and Applied Logic 42 (2):125-163.

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