CCC forcing and splitting reals

Boban Velickovic
2005 Israel Journal of Mathematics  
Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the
more » ... roper Forcing Axiom and is proved using the P -ideal dichotomy first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. In the process,
doi:10.1007/bf02785365 fatcat:hqvowqwmgrgstfdf6wfb6t6rua