## Notre Dame Journal of Formal Logic

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

### Rudimentary Recursion, Gentle Functions and Provident Sets

A. R. D. Mathias and N. J. Bowler

#### 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 ${\Delta}_{1}$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 24 March 2015

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00294527-2835101

**Mathematical Reviews number (MathSciNet)**

MR3326588

**Zentralblatt MATH identifier**

1371.03061

**Subjects**

Primary: 03E30: Axiomatics of classical set theory and its fragments 03D65: Higher-type and set recursion theory

Secondary: 03E40: Other aspects of forcing and Boolean-valued models 03E45: Inner models, including constructibility, ordinal definability, and core models

**Keywords**

rudimentary recursion provident set, provident closure gentle function attain delay progress (canonical, strict, solid)

#### 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