Abstract
Throughout, M will denote a transitive model of ZFC. Using the terms “random” and “generic” in the sense of [1], one may ask whether there can exist real numbers x and y such that x is generic over M[y] and y is random over M[x]. We shall see below by an elementary argument that this is not possible, and so, in a crude sense at least, random and generic extensions do not commute. This does not however rule out the possibility of a weaker commutativity. Let B be the complete Boolean algebra (in M) for adjoining a random real followed by a generic real and C be the complete Boolean algebra for adjoining a generic real followed by a random real. Then it still might be the case that B and C are isomorphic. This also fails, though, and we shall prove this by establishing the following combinatorial properties of MB and MC:butIn addition this will show that C cannot be embedded as a complete subalgebra of B.The property satisfied by B is reminiscent of calibre ℵ1 [2]. B would have calibre if we could replace “infinite” by “uncountable”, and this occurs if Martin's Axiom holds in M. To obtain the nonisomorphism of B and C in general necessitated looking at the weaker property.
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