Saturation properties of ideals in generic extensions. II
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- by James E. Baumgartner and Alan D. Taylor
- Trans. Amer. Math. Soc. 271 (1982), 587-609
- DOI: https://doi.org/10.1090/S0002-9947-1982-0654852-4
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Abstract:
The general type of problem considered here is the following. Suppose $I$ is a countably complete ideal on ${\omega _1}$ satisfying some fairly strong saturation requirement (e.g. $I$ is precipitous or ${\omega _2}$-saturated), and suppose that $P$ is a partial ordering satisfying some kind of chain condition requirement (e.g. $P$ has the c.c.c. or forcing with $P$ preserves ${\omega _1}$). Does it follow that forcing with $P$ preserves the saturation property of $I$? In this context we consider not only precipitous and ${\omega _2}$-saturated ideals, but we also introduce and study a class of ideals that are characterized by a property lying strictly between these two notions. Some generalized versions of Chang’s conjecture and Kurepa’s hypothesis also arise naturally from these considerations.References
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Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 271 (1982), 587-609
- MSC: Primary 03C62; Secondary 03E05, 03E35, 03E40, 03E55
- DOI: https://doi.org/10.1090/S0002-9947-1982-0654852-4
- MathSciNet review: 654852