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

by Shelah. [Sh:480]

Israel J Math, 1994

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.

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