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arXiv · 2507.11643

Notes on the equiconsistency of ZFC without the Power Set axiom and second order PA

Vladimir Kanovei·Vassily Lyubetsky
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Abstract

We demonstrate that theories $\text{Z}^-$, $\text{ZF}^-$, $\text{ZFC}^-$ (minus means the absence of the Power Set axiom) and $\text{PA}_2$, $\text{PA}_2^-$ (minus means the absence of the Countable Choice schema) are equiconsistent to each other. The methods used include the interpretation of a power-less set theory in $\text{PA}_2^-$ via well-founded trees, as well as the G\"odel constructibility in the said power-less set theory.

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Vladimir Kanovei, Vassily Lyubetsky. 2025-07-15. Notes on the equiconsistency of ZFC without the Power Set axiom and second order PA. https://arxiv.org/abs/2507.11643

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