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… Expand

It is shown that there is not a similar result for subsets of ω 2 L, and a number of related problems are considered, examining the extent to which they are “solvable” in the above sense.Expand

It is shown, assuming that 0# exists, that such models necessarily contain Mahlo cardinals of high order, but without further assumptions need not contain a cardinal κ which is κ-Mahlo.Expand

Set theory entered the modern era through the work of Godel and Cohen. This work provided set-theorists with the necessary tools to analyse a large number of mathematical problems which are… Expand

In this paper we isolate the notion of Stratified class forcing and show that Stratification implies cofinality-preservation and is preserved by iterations with the appropriate support. Many familiar… Expand