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VON NEUMANN'S PROBLEM AND LARGE CARDINALS

Published online by Cambridge University Press:  19 December 2006

ILIJAS FARAH  and
BOBAN VELIČKOVIĆ
ILIJAS FARAH
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3 and Matematicki Institut, Kneza Mihaila 35, Beograd, Serbia and [email protected]
BOBAN VELIČKOVIĆ
Affiliation:
Equipe de Logique Mathématique, UFR de Mathématiques (case 7012), Université Denis-Diderot Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, [email protected]
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Abstract

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.

Keywords

03E5528A60 (primary)03E75 (secondary)
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 38 , Issue 6 , December 2006 , pp. 907 - 912
Copyright
The London Mathematical Society 2006

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