Needed reals and recursion in generic reals Original Research Article

We consider sets of reals that are “adequate” in various senses, for example dominating or unbounded or splitting or non-meager. Call a real x “needed” (in any of these senses) if every adequate set contains a real in which x is recursive. We characterize the needed reals for numerous senses of “adequate.” We also consider, for various notions of forcing that add reals, the problem of characterizing the ground-model reals that are recursive in generic reals.