TY - JOUR
T1 - Complete quotient boolean algebras
AU - Kanamori, Akihiro
AU - Shelah, Saharon
PY - 1995/6
Y1 - 1995/6
N2 - 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.).
AB - 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.).
UR - http://www.scopus.com/inward/record.url?scp=0039913784&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-1995-1282888-0
DO - 10.1090/S0002-9947-1995-1282888-0
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AN - SCOPUS:0039913784
SN - 0002-9947
VL - 347
SP - 1963
EP - 1979
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 6
ER -