A study of elementary inductive definitions (e.i.d.) in HA. Strictly positive e.i.d. have closure ordinals = or < omega, and define predicates that are already definable in HA. The authors enlarge the class by adding so-called J-operators. E.i.d. in the larger class have closure ordinals up to omega + omega, but the authors are conservative over HA with respect to definability.
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