{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:03:32Z","timestamp":1773446612621,"version":"3.50.1"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T00:00:00Z","timestamp":1598313600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Zermelo\u2019s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle.<\/jats:p>","DOI":"10.1017\/s1755020320000301","type":"journal-article","created":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T03:40:48Z","timestamp":1598326848000},"page":"339-350","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["A NOTE ON CHOICE PRINCIPLES IN SECOND-ORDER LOGIC"],"prefix":"10.1017","volume":"16","author":[{"given":"BENJAMIN","family":"SISKIND","sequence":"first","affiliation":[]},{"given":"PAOLO","family":"MANCOSU","sequence":"additional","affiliation":[]},{"given":"STEWART","family":"SHAPIRO","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,8,25]]},"reference":[{"key":"S1755020320000301_r3","unstructured":"Gassner, C. (1985). Das Auswahlaxiom im Pr\u00e4dikatenkalk\u00fcl zweiter Stufe. 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