{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T22:40:49Z","timestamp":1649112049133},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":21651,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1954,12]]},"abstract":"<jats:p>In this paper a theorem will be established which states that a particular axiom of choice is equivalent to complete distributivity of union and intersection. The theorem will be formulated and proved in the system of logic of [4]. In addition to definitions of [4], the following will be used.<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200086655_eqn1\" \/><\/jats:disp-formula><\/jats:p><jats:p>In terms of these definitions, the theorem can be formulated as follows.<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200086655_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>The dual of this statement, obtained by interchanging <jats:italic>I<\/jats:italic> and <jats:italic>U<\/jats:italic>, is also a theorem and has a similar proof.<\/jats:p>","DOI":"10.2307\/2267734","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:34:32Z","timestamp":1146944072000},"page":"275-277","source":"Crossref","is-referenced-by-count":3,"title":["Distributivity and an axiom of choice"],"prefix":"10.1017","volume":"19","author":[{"given":"George E.","family":"Collins","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200086655_ref005","first-page":"3","volume-title":"Treatise on set topology","year":"1947"},{"key":"S0022481200086655_ref004","volume-title":"Mathematical logic","year":"1951"},{"key":"S0022481200086655_ref001","first-page":"146","volume-title":"Lattice theory","volume":"25","year":"1948"},{"key":"S0022481200086655_ref002","first-page":"10","volume-title":"Reele Funktionen","year":"1932"},{"key":"S0022481200086655_ref003","first-page":"19","volume-title":"Mengenlehre","year":"1935"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200086655","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T05:46:49Z","timestamp":1559886409000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200086655\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1954,12]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1954,12]]}},"alternative-id":["S0022481200086655"],"URL":"https:\/\/doi.org\/10.2307\/2267734","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1954,12]]}}}