Complete quotient boolean algebras

Akihiro Kanamori, Saharon Shelah

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

For I a proper, countably complete ideal on the power set p(X) for some set X, can the quotient Boolean algebra p(X)/I be complete? We first show that, if the cardinality of X is at least É3, 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 k which is k+-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over k = É1relative 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: 2É1= É3and there is an ideal ideal I over É1suchthat p((É1)I is complete. (The cardinality assertion implies that there is no ideal over É1which is É2- saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.).

Original languageEnglish
Pages (from-to)1963-1979
Number of pages17
JournalTransactions of the American Mathematical Society
Volume347
Issue number6
DOIs
StatePublished - Jun 1995

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