## Notre Dame Journal of Formal Logic

### 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 $M\cup\{\mathcal{G}\}$. 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 $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 1 (2015), 3-60.

Dates
First available in Project Euclid: 24 March 2015

https://projecteuclid.org/euclid.ndjfl/1427202973

Digital Object Identifier
doi:10.1215/00294527-2835101

Mathematical Reviews number (MathSciNet)
MR3326588

Zentralblatt MATH identifier
1371.03061

#### Citation

Mathias, A. R. D.; Bowler, N. J. Rudimentary Recursion, Gentle Functions and Provident Sets. Notre Dame J. Formal Logic 56 (2015), no. 1, 3--60. doi:10.1215/00294527-2835101. https://projecteuclid.org/euclid.ndjfl/1427202973

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