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- W2951707516 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.)" @default.
- W2951707516 created "2019-06-27" @default.
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- W2951707516 date "1995-06-01" @default.
- W2951707516 modified "2023-09-25" @default.
- W2951707516 title "Complete Quotient Boolean Algebras" @default.
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- W2951707516 doi "https://doi.org/10.2307/2154916" @default.
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