### Non-Cohen Oracle c.c.c.

by Shelah. [Sh:669]

J Applied Analysis, 2006

The oracle c.c.c. is closed related to Cohen forcing. During
an
iteration we can ``omit a type''; i.e. preserve ``the intersection
of
a given family of Borel sets of reals is empty'' provided that
Cohen
forcing satisfies it. We generalize this to other cases. In section
1 we
replace Cohen by ``nicely'' definable c.c.c., do the parallel of the
oracle c.c.c. and end with a criterion for extracting a subforcing
(not a complete subforcing) of a given nicely one and satisfying the
oracle.

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