Generic Saturation

Assuming that ORD is ! + !-Erdd os we show that if a class forcing amenable to L (an L-forcing) has a generic then it has one deenable in a set-generic extension of LO # ]. In fact we may choose such a generic to be periodic in the sense that it preserve the indiscernibility of a nal segment of a periodic subclass of the Silver indiscernibles, and therefore to be almost codable in the sense that it is deenable from a real which is generic for an L-forcing (and which belongs to a set-generic extension of LO # ]). This result is best possible in the sense that for any countable ordinal there is an L-forcing which has generics but none periodic of period. However, we do not know if an assumption beyond ZFC + \O # exists" is actually necessary for these results. Let P denote a class forcing deenable over an amenable ground model hL;Ai and assume that O # exists.