Complete quotient Boolean algebras

Abstract

For I I a proper, countably complete ideal on the power set P ( x ) \mathcal {P}(x) for some set X X , can the quotient Boolean algebra P ( X ) / I \mathcal {P}(X)/I be complete? We first show that, if the cardinality of X X is at least ω 3 {\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 I is a (nontrivial) ideal over a cardinal κ \kappa which is κ + {\kappa ^ + } -saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over κ = ω 1 \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 ω 1 = ω 3 {2^{{\omega _1}}} = {\omega _3} and there is an ideal ideal I I over ω 1 {\omega _1} such that P ( ω 1 ) / I \mathcal {P}({\omega _1})/I is complete. (The cardinality assertion implies that there is no ideal over ω 1 {\omega _1} which is ω 2 {\omega _2} -saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)