Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic

Kanovei, Vladimir; Lyubetsky, Vassily

doi:10.3390/axioms14120865

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Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic

by

Vladimir Kanovei

^*,†

and

Vassily Lyubetsky

^*,†

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia

^*

Authors to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Axioms 2025, 14(12), 865; https://doi.org/10.3390/axioms14120865

Submission received: 18 October 2025 / Revised: 19 November 2025 / Accepted: 21 November 2025 / Published: 25 November 2025

(This article belongs to the Section Logic)

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Abstract

We demonstrate that theories Z⁻, ZF⁻, ZFC⁻ (minus means the absence of the Power Set axiom) and PA₂ , 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 PA⁻₂ via well-founded trees, as well as the Gödel constructibility in said power-less set theory.

Keywords: constructibility; theories without the PS axiom; second-order arithmetic; consistency

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MDPI and ACS Style

Kanovei, V.; Lyubetsky, V. Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic. Axioms 2025, 14, 865. https://doi.org/10.3390/axioms14120865

AMA Style

Kanovei V, Lyubetsky V. Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic. Axioms. 2025; 14(12):865. https://doi.org/10.3390/axioms14120865

Chicago/Turabian Style

Kanovei, Vladimir, and Vassily Lyubetsky. 2025. "Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic" Axioms 14, no. 12: 865. https://doi.org/10.3390/axioms14120865

APA Style

Kanovei, V., & Lyubetsky, V. (2025). Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic. Axioms, 14(12), 865. https://doi.org/10.3390/axioms14120865

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Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic

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