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Transactions of the American Mathematical Society

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



Complete quotient Boolean algebras

Authors: Akihiro Kanamori and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 347 (1995), 1963-1979
MSC: Primary 03E35; Secondary 03E40, 03E55, 06E05
MathSciNet review: 1282888
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Abstract: For $ I$ a proper, countably complete ideal on the power set $ \mathcal{P}(x)$ for some set $ X$, can the quotient Boolean algebra $ \mathcal{P}(X)/I$ be complete? We first show that, if the cardinality of $ X$ is at least $ {\omega _3}$, then having completeness implies the existence of an inner model with a measurable cardinal. A well-known situation that entails completeness is when the ideal $ I$ is a (nontrivial) ideal over a cardinal $ \kappa $ which is $ {\kappa ^ + }$-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over $ \kappa = {\omega _1}$ relative to the existence of a Woodin cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large cardinal hypothesis, the consistency of: $ {2^{{\omega _1}}} = {\omega _3}$ and there is an ideal ideal $ I$ over $ {\omega _1}$ such that $ \mathcal{P}({\omega _1})/I$ is complete. (The cardinality assertion implies that there is no ideal over $ {\omega _1}$ which is $ {\omega _2}$-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)

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