Complete quotient Boolean algebras

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.)