Generic saturation

Extract

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.

Definition. 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^#].

Remark. The reverse Easton product of Cohen forcings 2^<κ, κ regular is relevant. So are the Easton product and the full product, provided κ is restricted to the successor cardinals. See Chapter 3, Section Two of Friedman [3]. Of course any set-forcing (in L) is almost relevant.

Definition. κ is α-Erdös if whenever C is CUB in κ and f: [C]^<ω → κ is regressive (i.e., f(a) < min(a)) then f has a homogeneous set of ordertype α.