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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Saturation properties of ideals in generic extensions. II


Authors: James E. Baumgartner and Alan D. Taylor
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
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0654852-4
Article copyright: © Copyright 1982 American Mathematical Society