There is a family of questions in relativized complexity theory--weak analogs of the Friedberg Jump-Inversion Theorem--that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines $aw2$-generic sets. i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. $Aw2$-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any set $X$ there is an $aw2$-generic set $G$ such that $\mathbf{NP}^G \cap co-\mathbf{NP}^G \nsubseteq \mathbf{P}^{G \oplus X}$. (On the other hand, if $G$ is 1-generic, then $\mathbf{NP}^G \cap co-\mathbf{NP}^G \subseteq \mathbf{P}^{G \oplus \mathrm{SAT}}$. where $\mathrm{SAT}$ is the $\mathbf{NP}$-complete satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of $aw2$-generic sets differ radically from those of 1-generic sets: every degree above $O'$ contains an $aw2$-generic set: no $aw2$-generic set exists below any incomplete r.e. set; there is an $aw2$-generic set which is the join of two Turing equivalent $aw2$-generic sets. Finally, a result of Shore is presented [30] which states that every degree above $0'$ is the jump of an $aw2$-generic degree.