{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,28]],"date-time":"2026-06-28T09:47:12Z","timestamp":1782640032763,"version":"3.54.5"},"reference-count":42,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T00:00:00Z","timestamp":1764028800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Institute for Information Transmission Problems of the Russian Academy of Sciences"},{"DOI":"10.13039\/501100003443","name":"Ministry of Education and Science of the Russian Federation","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100003443","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We demonstrate that theories Z\u2212,\u00a0ZF\u2212,\u00a0ZFC\u2212 (minus means the absence of the Power Set axiom) and PA2, PA2\u2212 (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\u2212 via well-founded trees, as well as the G\u00f6del constructibility in said power-less set theory.<\/jats:p>","DOI":"10.3390\/axioms14120865","type":"journal-article","created":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T17:31:34Z","timestamp":1764091894000},"page":"865","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7415-9784","authenticated-orcid":false,"given":"Vladimir","family":"Kanovei","sequence":"first","affiliation":[{"name":"Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3739-9161","authenticated-orcid":false,"given":"Vassily","family":"Lyubetsky","sequence":"additional","affiliation":[{"name":"Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"321","DOI":"10.2307\/2270324","article-title":"A survey of proof theory","volume":"33","author":"Kreisel","year":"1968","journal-title":"J. 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