Von Neumann’s Problem and Large Cardinals

It is a well known problem of Von Neumann 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–Pazák and Velickovic 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, i.e., is a Maharam algebra. We use some ideas of Gitik and Shelah to show that some large cardinal assumptions are necessary for this result. In 1937 von Neumann asked whether every ccc weakly distributive complete Boolean algebra is a measure algebra ([10]). We show that a positive answer to von Neumann’s problem, if consistent at all, requires a large cardinal assumption. A complete Boolean algebra is a Maharam algebra if it carries a strictly positive continuous submeasure ([13]). Every measure algebra is a Maharam algebra, and every Maharam algebra has ccc and is weakly distributive (see e.g., [3]). Theorem 1. Assume every ccc weakly distributive complete Boolean algebra is a Maharam algebra. Then there is an inner model with a measurable cardinal κ such that o(κ) = κ. By results of [13] and [1], a consequence of the Proper Forcing Axiom implies that every ccc, weakly distributive, complete Boolean algebra is a Maharam algebra. Our result gives a lower bound for the consistency strength of this statement and completes the answer to [4, Problem AU(d)]. The remaining part of von Neumann’s problem, whether every Maharam algebra is a measure algebra, is known under the names of Maharam’s Problem and Control Measure Problem (see [9], [7], [2, §393]). Theorem 2. Assume every weakly distributive complete Boolean algebra B such that every completely countably generated subalgebra is a measure algebra and B has property K is a Maharam algebra. Then there is an inner model with a measurable cardinal κ such that o(κ) = κ. Terminology. A subset of a Boolean algebra is an antichain if it consists of nonzero elements but the meet of any two of its members is zero. A Boolean algebra is ccc if it does not have uncountable antichaims. A complete Boolean algebra is weakly distributive if for every sequence An (n ∈ N) of maximal antichains there is a maximal antichain A such that for every a ∈ A and n ∈ N the set Date: December 22, 2004 Version of January 31, 2005. 1991 Mathematics Subject Classification. Primary: 03Exx, Secondary: 28Axx. We are indebted to Rich Laver for pointing out that [6] may be relevant to von Neumann’s problem and to Sy Friedman for suggesting Lemma 5. These results were obtained in December 2004 while we were visiting E. Schrödinger Institute in Vienna. We would like to thank the Institute for providing hospitality and a stimulating environment. Filename: n2004l22.tex.