{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T10:03:47Z","timestamp":1748253827719},"reference-count":4,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2018,7,4]],"date-time":"2018-07-04T00:00:00Z","timestamp":1530662400000},"content-version":"vor","delay-in-days":3,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2018,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>With the Axiom of Choice <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0001\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0002\" \/> for any infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0003\" \/> but, without <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0004\" \/>, we cannot conclude any relationship between <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0005\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0006.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0006\" \/> for an arbitrary infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0007.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0007\" \/>. In this paper, we give some properties of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0008.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0008\" \/> in the absence of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0009.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0009\" \/> and compare them to those of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0010.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0010\" \/> for an infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0011.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0011\" \/>. Among our results, we show that \u201c<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0012.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0012\" \/> for any infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0013.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0013\" \/> and any natural number <jats:italic>n<\/jats:italic>\u201d is provable in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0014.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0014\" \/> although \u201c<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0015.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0015\" \/> for any infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700054-math-0016.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700054:malq201700054-math-0016\" \/>\u201d is not.<\/jats:p>","DOI":"10.1002\/malq.201700054","type":"journal-article","created":{"date-parts":[[2018,7,5]],"date-time":"2018-07-05T05:30:18Z","timestamp":1530768618000},"page":"201-206","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Some properties of infinite factorials"],"prefix":"10.1002","volume":"64","author":[{"given":"Nattapon","family":"Sonpanow","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science Chulalongkorn University Payathai Road Patumwan Bangkok 10330 Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pimpen","family":"Vejjajiva","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science Chulalongkorn University Payathai Road Patumwan Bangkok 10330 Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2018,7,4]]},"reference":[{"key":"e_1_2_5_2_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-93-3-185-195"},{"key":"e_1_2_5_3_1","volume-title":"The Axiom of Choice, Studies in Logic and the Foundations of Mathematics","author":"Jech T. J.","year":"1973"},{"key":"e_1_2_5_4_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-98-1-23-24"},{"key":"e_1_2_5_5_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19570031302"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201700054","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201700054","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T19:25:26Z","timestamp":1694978726000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201700054"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,7]]}},"alternative-id":["10.1002\/malq.201700054"],"URL":"https:\/\/doi.org\/10.1002\/malq.201700054","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7]]},"assertion":[{"value":"2017-09-02","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2017-11-14","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2018-07-04","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}