Rudimentary Recursion, Gentle Functions and Provident Sets

Notre Dame Journal of Formal Logic 56 (1):3-60 (2015)
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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 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 Shoenfield’s unramified forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and 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 $\Delta_{1}$



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Constructing the constructible universe constructively.Michael Rathjen - 2024 - Annals of Pure and Applied Logic 175 (3):103392.
Infinite Computations with Random Oracles.Merlin Carl & Philipp Schlicht - 2017 - Notre Dame Journal of Formal Logic 58 (2):249-270.

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Reconsidering ordered pairs.Dana Scott & Dominic McCarty - 2008 - Bulletin of Symbolic Logic 14 (3):379-397.
A converse of the Barwise completeness theorem.Jonathan Stavi - 1973 - Journal of Symbolic Logic 38 (4):594-612.

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