CCC forcing and splitting reals |
| |
Authors: | Boban Velickovic |
| |
Institution: | (1) Equipe de Logique, Université de Paris 7, 2 Place Jussieu, 75251 Paris, France |
| |
Abstract: | The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms
of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive
posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly
distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This
is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic AT] and later extended by Todorcevic T].
As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra
is a Maharam algebra. Using a previous theorem of Shelah Sh1] it also follows that a modified Prikry conjecture holds in
the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open
algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set
of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the
result of AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc
forcing adds a splitting real. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|