Complete quotient Boolean algebras
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- by Akihiro Kanamori and Saharon Shelah PDF
- Trans. Amer. Math. Soc. 347 (1995), 1963-1979 Request permission
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.)References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 1963-1979
- MSC: Primary 03E35; Secondary 03E40, 03E55, 06E05
- DOI: https://doi.org/10.1090/S0002-9947-1995-1282888-0
- MathSciNet review: 1282888