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

DOI:
https://doi.org/10.1090/S0002-9947-1995-1282888-0

MathSciNet review:
1282888

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Abstract: For a proper, countably complete ideal on the power set for some set , can the quotient Boolean algebra be complete? We first show that, if the cardinality of is at least , 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 is a (nontrivial) ideal over a cardinal which is -saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over 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: and there is an ideal ideal over such that is complete. (The cardinality assertion implies that there is no ideal over which is -saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1282888-0

Article copyright:
© Copyright 1995
American Mathematical Society