Inherent enumerability of strong jump-traceability

We show that every strongly jump-traceable set obeys every benign cost function. Moreover, we show that every strongly jump-traceable set is computable from a computably enumerable strongly jump-traceable set. This allows us to generalise properties of c.e.\ strongly jump-traceable sets to all such sets. For example, the strongly jump-traceable sets induce an ideal in the Turing degrees; the strongly jump-traceable sets are precisely those that are computable from all superlow Martin-L"{o}f random sets; the strongly jump-traceable sets are precisely those that are a base for ${\text{Demuth}}_{\text{BLR}}$-randomness; and strong jump-traceability is equivalent to strong superlowness.