The Genericity Conjecture

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〉. Thus Theorem A is established by producing a real R s.t. O / ∈ L[R] yet Sat〈L,A〉 is definable over 〈L[R], A〉 for each L-amenable A. A weaker version of the Genericity Conjecture would state: If O / ∈ L[R] then either R ∈ L or R is generic over some inner model M not containing R. This version of the conjecture is still open. However, this question can also be studied in contexts where O does not exist, for example when the universe has ordinal height equal to that of the minimal transitive model of ZF. In the latter context, Mack Stanley [93] has demonstrated the consistency of the existence of a non-constructible real which belongs to every inner model over which it is generic.