A study of elementary inductive definitions (e.i.d.) in HA. Strictly positive e.i.d. have closure ordinals ≤ω, and define predicates that are already definable in HA. We enlarge this class by adding so-called J-operators, for example ⌝ ⌝. E.i.d. in this larger class have closure ordinals up to ω+ω, but they are conservative over HA w.r.t. definability.