Sets Having Calibre

Abstract

Publisher Summary It is well-known that if the complete Boolean algebra satisfies the countable chain condition (c.c.c.) then cardinals are preserved in a Boolean extension(similarly for forcing, using a partially ordered set with greatest element). However the c.c.c. has its disadvantages, the main from the point of view being that the Boolean algebra may satisfy the c.c.c. in V without satisfying it in the Boolean extension. This occurs for example if Boolean algebra is a Souslin algebra. Thus, one leads to consider stronger notions, which have more satisfactory closure and extension properties. The chapter discusses the general properties of sets having caliber. A typical such property is that if Boolean algebra has calibre then it still has caliber in any c.c.c. extension. The chapter is concerned with certain specific Boolean algebras, notably the “amoeba” and “dominating” algebras, which are good examples of algebras having caliber, and which have considerable importance for the study of Lebesgue measure and Baire category on the real line.