The oracle c.c.c. is closely 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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