Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. 1.

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Abstract. An informal sketch (with intermittent details) of parts of E-Recursion theory, mostly old, some new, that stresses intuition. The lack of effective unbounded search is balanced by the availability of divergence witnesses. A set is E-closed iff it is transitive and closed under the application of partial E-recursive functions. Some finite injury, forcing, and model theoretic constructions can be adapted to E-closed sets that are not Σ1 admissible.