Series: Penn State Logic Seminar Date: Tuesday, April 24, 2001 Time: 2:30 - 3:20 PM Place: 316 Willard Building Speaker: Natasha Dobrinen, Mathematics, University of Minnesota Title: Complete embeddings of the Cohen algebra into three classic examples of complete, non-measurable, atomless, c.c.c. Boolean algebras. Abstract: Von Neumann conjectured that every complete, c.c.c. Boolean algebra which satisfies the weak (omega,omega)-distributive law carries a strictly positive, sigma-additive measure. Although consistent counterexamples have been obtained, whether von Neumann's conjecture is consistent with ZFC remains an open problem. In view of this, it is of interest to investigate distributive laws in complete, c.c.c. Boolean algebras. In this talk, I will construct complete embeddings of the Cohen algebra into several classic examples of complete, non-measurable, atomless, c.c.c. Boolean algebras, namely, the Gaifman, Argyros, and Galvin-Hajnal algebras. Since the Cohen algebra does not satisfy any form of distributivity, the complete embedding of the Cohen algebra into each of these three Boolean algebras implies that they are completely non-distributive. This leads to the question: Within ZFC, is there a complete, non-measurable, atomless, c.c.c. Boolean algebra into which the Cohen algebra does not embed as a complete subalgebra?
Mathematical Logic at Penn State
Department of Mathematics