How special are Cohen and random forcings i.e. Boolean algebras of the family of subsets of reals modulo meagre or null

Abstract

The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or ``minimal'' or at least characteristic in a family of ``nice forcing'' like Borel. We shall interpret ``nice'' as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among the rest random is quite characteristic even unique. Concerning the second class we have weak results, concerning the first class, our results look satisfactory. We have two main results: one (1.14) says that Cohen forcing is ``minimal'' in the first class, the other (1.10) says that all c.c.c. Souslin forcing have a property shared by Cohen forcing and Random real forcing, so it gives a weak answer to the problem on how special is random forcing, but says much on all c.c.c. Souslin forcing.