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Von Neumann's Problem and Large Cardinals
Author(s) -
Farah Ilijas,
Veličković Boban
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018704
Subject(s) - mathematics , distributive property , complete boolean algebra , modulo , countable set , distributivity , boolean algebra , von neumann algebra , mathematics subject classification , stone's representation theorem for boolean algebras , von neumann architecture , discrete mathematics , two element boolean algebra , regular cardinal , consistency (knowledge bases) , pure mathematics , algebra over a field , algebra representation
It is a well‐known problem of Von Neumann to discover whether the countable chain condition and weak distributivity of a complete Boolean algebra imply that it carries a strictly positive probability measure. It was shown recently by Balcar, Jech and Pazák, and by Veličković, that it is consistent with ZFC, modulo the consistency of a supercompact cardinal, that every ccc weakly distributive complete Boolean algebra carries a contiuous strictly positive submeasure – that is, it is a Maharam algebra. We use some ideas of Gitik and Shelah and implications from the inner model theory to show that some large cardinal assumptions are necessary for this result. 2000 Mathematics Subject Classification 03E55, 28A60 (primary), 03E75 (secondary).