Complete Embeddings of the Cohen Algebra into Three Families of C.c.c., Non-measurable Boolean Algebras
نویسندگان
چکیده
Von Neumann conjectured that the countable chain condition and the weak (ω, ω)-distributive law characterize measurable algebras among Boolean σalgebras [Mau]. Consistent counter-examples have been obtained by Maharam [Mah], Jensen [J], Glówczyński [Gl], and Veličković [V]. However, whether von Neumann’s proposed characterization of measurable algebras fails within ZFC remains an open problem. In searching for possible counter-examples to von Neumann’s proposed characterization of measurable algebras within ZFC, we investigated three families of complete, c.c.c., non-measurable Boolean algebras, namely, the Argyros, Galvin-Hajnal, and Gaifman algebras, to find out whether these Boolean algebras sustain any weak form of distributivity. By the Cohen algebra we mean the completion of any countable, atomless Boolean algebra. By the κ-Cohen algebra we mean the completion of the free Boolean algebra on κ-many generators. We found the following:
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