Generic saturation

Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to L (an L-forcing) has a generic then it has one definable in a set-generic extension of L [ O # ]. In fact we may choose such a generic to be periodic in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be almost codable in the sense that it is definable from a real which is generic for an L -forcing (and which belongs to a set-generic extension of L [ 0 # ]). This result is best possible in the sense that for any countable ordinal α there is an L -forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+“ O # exists” is actually necessary for these results. Let P denote a class forcing definable over an amenable ground model 〈 L , A 〉 and assume that O # exists. D efinition . P is relevant if P has a generic definable in L [ 0 # ]. P is almost relevant if P has a generic definable in a set-generic extension of L [ 0 # ]. R emark . The reverse Easton product of Cohen forcings 2