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Saturation Properties of Ideals in Generic Extensions.II

James E. Baumgartner and Alan D. Taylor
Transactions of the American Mathematical Society
Vol. 271, No. 2 (Jun., 1982), pp. 587-609
DOI: 10.2307/1998900
Stable URL: http://www.jstor.org/stable/1998900
Page Count: 23
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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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