On Weak Set Theories Interpreted in PA

Junhong Chen

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arXiv · 2608.16685

On Weak Set Theories Interpreted in PA

Junhong Chen

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Abstract

We classify a broad family of weak first-order set theories, under ordinary parameter-free interpretability, by the first-order arithmetical theories with which they are mutually interpretable. This also determines their consistency strength. Set theories that correspond to the same arithmetical theory are often related by deductive extension. It is therefore enough to interpret a stronger set theory in the arithmetical theory and to recover the arithmetical theory in a weaker set theory; all intermediate cases then follow. The set theories under consideration fall roughly into three classes: theories with neither Power Set nor Infinity, theories with Power Set but without Infinity, and theories with Infinity but without Power Set. We conclude with a brief account of the higher levels that remain to be investigated.

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Junhong Chen. 2026-08-17. On Weak Set Theories Interpreted in PA. https://arxiv.org/abs/2608.16685

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