Notre Dame Journal of Formal Logic

Rudimentary Recursion, Gentle Functions and Provident Sets

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

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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 Jω. 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{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 Δ˙0 formulæ is Δ1.

Article information

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

First available in Project Euclid: 24 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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