The genericity conjecture

The Genericity Conjecture , as stated in Beller-Jensen-Welch [1], is the following: (*) If O # ∉ L[R], R ⊆ ω , then R is generic over L . We must be precise about what is meant by “generic”. Definition (Stated in Class Theory). A generic extension of an inner model M is an inner model M[G] such that for some forcing notion ⊆ M : (a) 〈 M, 〉 is amenable and ⊩ is 〈 M, 〉-definable for sentences. (b) G ⊆ is compatible, closed upwards, and intersects every 〈 M , 〉-definable dense D ⊆ . A set x is generic over M if it is an element of a generic extension of M . And x is strictly generic over M if M[x] is a generic extension of M . Though the above definition quantifies over classes, in the special case where M = L and O # exists, these notions are in fact first order, as all L -amenable classes are definable over L[O # ]. From now on assume that O # exists. Theorem A. The Genericity Conjecture is false . The proof is based upon the fact that every real generic over L obeys a certain definability property, expressed as follows. Fact. If R is generic over L, then for some L-amenable class A , Sat〈 L, A 〉 is not definable over 〈 L [ R ], A 〉, where Sat〈 L,A 〉 is the canonical satisfaction predicate for 〈 L,A 〉.