Invisible Genericity and 0#

0 # can be invisibly class generic. 1. Introduction Roughly, the main theorem of this paper is that some instances of any type of non-constructible object are class generic over L, in one sense. Since small large cardinal properties are inherited in L, let us begin by considering the sense in which 0 # can be generic. It is well known that 0 # is not set generic over L, or, indeed, over any inner model M such that 0 # / ∈ M. This is because, if 0 # / ∈ M , then unboundedly many M-cardinals are collapsed in L[0 # ], a trick no set forcing can perform. It is also known that there can be no L-definable class forcing P such that 0 # ∈ L[G] and L[G]; G ZFC, for some L-definably generic G ⊆ P. This is because it is known that in such a case the forcing relation on a cone of conditions rooted in G must be L-definable because L[G]; L, P, G ZFC; but then ∃p ∃ι p P " ˇ ι is an indiscernible " and L ι ϕ defines L-truth over L. Further, it is known that there is no L-amenable class forcing property P such that 0 # ∈ L[G] and L[G]; P, G ZFC, for some L; P-definably generic G ⊆ P. In such a case, again, the forcing relation on a cone of conditions rooted in G must be definable over L; P. Since P is L-amenable (and compatible with 0 #), the class P is definable over L[0 # ], and, in fact, the members of a tail of the indiscernibles are indiscernible I am grateful to Sy Friedman for pointing out a mistake in the first version, and to David Cook for pointing out an improvement regarding the definability of the forcing relation.