{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:51:14Z","timestamp":1759146674651},"reference-count":14,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2008,11,4]],"date-time":"2008-11-04T00:00:00Z","timestamp":1225756800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2008,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study statements about countable and well\u2010ordered unions and their relation to each other and to countable and well\u2010ordered forms of the axiom of choice. Using WO as an abbreviation for \u201cwell\u2010orderable\u201d, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union of countable sets is WO. (\u00a9 2008 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200710073","type":"journal-article","created":{"date-parts":[[2008,11,4]],"date-time":"2008-11-04T13:53:03Z","timestamp":1225806783000},"page":"652-665","source":"Crossref","is-referenced-by-count":6,"title":["Unions and the axiom of choice"],"prefix":"10.1002","volume":"54","author":[{"given":"Omar De la","family":"Cruz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eric J.","family":"Hall","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul","family":"Howard","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kyriakos","family":"Keremedis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean E.","family":"Rubin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2008,11,4]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.2307\/1999384"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310049"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/18.6.580"},{"key":"e_1_2_1_5_2","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1090\/S0002-9939-1964-0167517-7","article-title":"The index problem for infinite symmetric groups","volume":"15","author":"Gaughan E. D.","year":"1964","journal-title":"Proc. Amer. Math. Soc."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093636102"},{"key":"e_1_2_1_7_2","doi-asserted-by":"crossref","unstructured":"P.Howard andJ. E.Rubin Consequences of the Axiom of Choice. AMS Mathematical Surveys and Monographs (Amer. Math. Soc. 1998). Availlable athttp:\/\/consequences.emich.edu\/.","DOI":"10.1090\/surv\/059"},{"key":"e_1_2_1_8_2","unstructured":"T. J.Jech The Axiom of Choice (North\u2010Holland 1973)."},{"key":"e_1_2_1_9_2","unstructured":"T. J.Jech Set Theory The third millennium edition revised and expanded (Springer 2003)."},{"key":"e_1_2_1_10_2","first-page":"297","article-title":"On \u0398\u2010model of the set theory","volume":"14","author":"Jech T. J.","year":"1966","journal-title":"Bull. Acad. Polon. Sci. S\u00e9r. Sci. Math. Astronom. Phys."},{"key":"e_1_2_1_11_2","first-page":"351","article-title":"Applications of the \u0398\u2010model","volume":"14","author":"Jech T. J.","year":"1966","journal-title":"Bull. Acad. Polon. Sci. S\u00e9r. Sci. Math. Astronom. Phys."},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.2307\/2272420"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90011-0"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(89)90212-3"},{"key":"e_1_2_1_15_2","unstructured":"H.Rubin andJ. E.Rubin Equivalents of the Axiom of Choice II (North\u2010Holland 1985)."}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200710073","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200710073","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.200710073","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T13:48:00Z","timestamp":1699969680000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.200710073"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,11,4]]},"references-count":14,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2008,12]]}},"alternative-id":["10.1002\/malq.200710073"],"URL":"https:\/\/doi.org\/10.1002\/malq.200710073","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,11,4]]}}}