Rudimentary Recursion, Gentle Functions and Provident Sets

Abstract

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 $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega }$. $\mathsf{PROVI}$ 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 $\mathcal{G}$ is the provident closure of $M\cup \left\{\mathcal{G}\right\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for ${\dot{\Delta }}_0$ formulæ is ${\Delta }_1$.