This analysis shows equiconsistency between ZFC without the Power Set axiom and second order PA models, indicating important relationships among set theories.
We demonstrate that theories Z⁻, ZF⁻, ZFC⁻ (minus means the absence of the Power Set axiom) and PA₂, PA₂⁻ (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 PA₂⁻ via well-founded trees, as well as the Gödel constructibility in the said power-less set theory.
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Kanovei et al. (2025) studied this question.
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