Classes of barren extensions

Journal of Symbolic Logic 86 (1):178-209 (2021)
  Copy   BIBTEX

Abstract

Henle, Mathias, and Woodin proved in [21] that, provided that ${\omega }{\rightarrow }^{{\omega }}$ holds in a model M of ZF, then forcing with $$ 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 $\mathcal {P}/{\mathrm {Fin}}^{\otimes {\alpha }}$, $2\le {\alpha }<{\omega }_1$, forcing non-p-points also produce barren extensions.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 76,264

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Sets and classes as many.John L. Bell - 2000 - Journal of Philosophical Logic 29 (6):585-601.
Mass Nouns in a Logic of Classes as Many.Nino B. Cocchiarella - 2009 - Journal of Philosophical Logic 38 (3):343-361.
Definable Operators on Stable Set Lattices.Robert Goldblatt - 2020 - Studia Logica 108 (6):1263-1280.
Glivenko sequent classes in the light of structural proof theory.Sara Negri - 2016 - Archive for Mathematical Logic 55 (3-4):461-473.
Cercidas, Frag. 2, ii. 12.Arthur Platt - 1912 - Classical Quarterly 6 (01):43-.
Bimodal logics for extensions of arithmetical theories.Lev D. Beklemishev - 1996 - Journal of Symbolic Logic 61 (1):91-124.
Topics in Lukasiewicz Logics.Merwin Gordon Beavers - 1991 - Dissertation, Indiana University
Models for relevant modal logics.André Fuhrmann - 1990 - Studia Logica 49 (4):501 - 514.

Analytics

Added to PP
2020-10-06

Downloads
2 (#1,401,720)

6 months
1 (#449,844)

Historical graph of downloads
How can I increase my downloads?