The primitive recursive functions were generalized to sets by Ronald Jensen and Carol Karp in their 1971 paper. Other higher order computability settings and models, in particular E-recursion and infinite time computation models, were later introduced by Dag Normann, Gerald Sacks, J.D. Hamkins, et alii. Various closure and reachability properties emerge from these settings and models, giving rise to the identification and study of specific ordinals linked to computability results and variants of the Lévy-Shoenfield absoluteness theorem. (LP version of my ASL 2018 special session talk)