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2015 Rudimentary Recursion, Gentle Functions and Provident Sets
A. R. D. Mathias, N. J. Bowler
Notre Dame J. Formal Logic 56(1): 3-60 (2015). DOI: 10.1215/00294527-2835101


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.


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A. R. D. Mathias. N. J. Bowler. "Rudimentary Recursion, Gentle Functions and Provident Sets." Notre Dame J. Formal Logic 56 (1) 3 - 60, 2015.


Published: 2015
First available in Project Euclid: 24 March 2015

zbMATH: 1371.03061
MathSciNet: MR3326588
Digital Object Identifier: 10.1215/00294527-2835101

Primary: 03D65 , 03E30
Secondary: 03E40 , 03E45

Keywords: attain , Delay , gentle function , progress (canonical, strict, solid) , provident set, provident closure , rudimentary recursion

Rights: Copyright © 2015 University of Notre Dame


Vol.56 • No. 1 • 2015
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