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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