Generic Saturation

Abstract

Assuming that ORD is $\omega +\omega $-Erd\"os 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 {\it 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 {\it 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 $L[O^\#]$).