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CLASSES OF BARREN EXTENSIONS

Part of: Extremal combinatorics Graph theory Model theory Set theory

Published online by Cambridge University Press:  05 October 2020

NATASHA DOBRINEN  and
DAN HATHAWAY
NATASHA DOBRINEN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF DENVER C.M. KNUDSON HALL ROOM 300-2390 S. YORK STREET DENVER, CO80208, USAE-mail: Natasha.Dobrinen@du.eduURL: http://web.cs.du.edu/~ndobrine
DAN HATHAWAY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF VERMONT INNOVATION HALL 82 UNIVERSITY PLACE BURLINGTON, VT05405, USAE-mail: Daniel.Hathaway@uvm.eduURL: https://www.uvm.edu/cems/mathstat/profiles/daniel-hathaway
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Abstract

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.

Keywords

ultrafiltersRamsey theoryforcingmodels without full choice

MSC classification

Primary: 03E02: Partition relations
Secondary: 03E05: Other combinatorial set theory 03C15: Denumerable structures 03E25: Axiom of choice and related propositions 05C55: Generalized Ramsey theory 05D10: Ramsey theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 178 - 209
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Copyright
© The Association for Symbolic Logic 2020

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