This paper, a contribution to ``micro set theory{''}, is the study promised by the first author in {[}M4], as improved and extended by work of the second. We use the rudimentarily recursive (set-theoretic) functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of PROVI, a subsystem of KP whose minimal model is Jensen's J(omega). PROVI supports familiar definitions, such as rank, transitive closure and ordinal addition-though not ordinal multiplication-and (shown in {[}M8]) Shoenfield's unramified forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and (shown in {[}M8]) the extension of a provident M by a set-generic G is the provident closure of M boolean OR \{G\}. The improvidence of many models of Z is shown. The final section uses similar but simpler recursions to show, in the weak system MW, that the truth predicate for (Delta) over dot(0) formulae is Delta(1).