Outer models and genericity

1. Introduction Why is forcing the only known method for constructing outer models of set theory? If V is a standard transitive model of ZFC, then a standard transitive model W of ZFC is an outer model of V if V ⊆ W and V ∩ OR = W ∩ OR. Is every outer model of a given model a generic extension? At one point Solovay conjectured that if 0 # exists, then every real that does not construct 0 # lies in L[G], for some G that is generic for some forcing P ∈ L. Famously, this was refuted by Jensen's coding theorem. He produced a real that is generic for an L-definable class forcing property, but does not lie in any set forcing extension of L. Beller, Jensen, and Welch in Coding the universe [BJW] revived Solovay's conjecture by asking the following question: Let a ⊆ ω be such that L[a] " 0 # does not exist ". Is there a b ∈ L[a] such that a / ∈ L[b] and a is set generic over L[b]. In [S1] it was shown that even if arbitrary inner models are allowed, rather than just ones of the form L[b], and even if we allow a to be class generic, the answer is No in general: Theorem 1.1. Let L α be a minimal countable standard transitive model of ZFC. There exists a real x nwg having the following three properties: (1) x nwg / ∈ L α. (2) L α [x nwg ] ZFC. (3) x nwg is not definably generic over any outer model of L α that does not already contain x nwg. A precise statement of (3) is the following: Assume that V is an outer model of L α and that P is a V-amenable partial ordering such that (V ; P) satisfies ZFC. Assume that the forcing relation restricted to sentences of bounded complexity is definable over (V ; P). (See Remark 1.8 regarding this hypothesis.) If G is a maximal filter on P meeting every dense subclass of P that is definable over (V ; P) and x nwg ∈ V [G], then x nwg ∈ V. For the sake of clarity, an elementary remark is in order. As is customary, we write " V " for the standard structure (V ; ∈). If S ⊆ V , then " (V ; S) …