{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T15:27:14Z","timestamp":1759073234750},"reference-count":7,"publisher":"Wiley","issue":"4-5","license":[{"start":{"date-parts":[[2016,7,12]],"date-time":"2016-07-12T00:00:00Z","timestamp":1468281600000},"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":[[2016,8]]},"abstract":"<jats:p>We show that (i) it is consistent with <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300054-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300054:malq201300054-math-0003\" \/> that there are infinite sets <jats:italic>X<\/jats:italic> on which every metric is discrete; (ii) the notion of <jats:italic>real infinite<\/jats:italic> is strictly stronger than that of <jats:italic>metrically infinite<\/jats:italic>; (iii) a set <jats:italic>X<\/jats:italic> is metrically infinite if and only if it is weakly Dedekind\u2010infinite if and only if the cardinality of the set of all metrically finite subsets of <jats:italic>X<\/jats:italic> is strictly less than the size of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300054-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300054:malq201300054-math-0004\" \/>; and (iv) an infinite set <jats:italic>X<\/jats:italic> is weakly Dedekind\u2010infinite if and only if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300054-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300054:malq201300054-math-0005\" \/> has infinite towers if and only if <jats:italic>X<\/jats:italic> has countable partitions.<\/jats:p>","DOI":"10.1002\/malq.201300054","type":"journal-article","created":{"date-parts":[[2016,7,12]],"date-time":"2016-07-12T07:01:54Z","timestamp":1468306914000},"page":"383-390","source":"Crossref","is-referenced-by-count":2,"title":["Non\u2010discrete metrics in  and some notions of finiteness"],"prefix":"10.1002","volume":"62","author":[{"given":"Kyriakos","family":"Keremedis","sequence":"first","affiliation":[{"name":"Department of Mathematics University of the Aegean Karlovassi 83200 Samos Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2016,7,12]]},"reference":[{"key":"e_1_2_3_2_1","doi-asserted-by":"publisher","DOI":"10.2307\/2275247"},{"key":"e_1_2_3_3_1","unstructured":"E.Hall K.Keremedis Independent Families and Some Notions of Finiteness preprint 2011."},{"key":"e_1_2_3_4_1","volume-title":"Mathematical Surveys and Monographs Vol. 59","author":"Howard P.","year":"1998"},{"issue":"3","key":"e_1_2_3_5_1","doi-asserted-by":"crossref","first-page":"169","DOI":"10.4064\/fm-133-3-169-177","article-title":"Definitions of finite","volume":"133","author":"Howard P.","year":"1989","journal-title":"Fundam. Math."},{"key":"e_1_2_3_6_1","volume-title":"Studies in Logic and the Foundations of Mathematics Vol. 75","author":"Jech T.","year":"1973"},{"key":"e_1_2_3_7_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4064\/fm-46-1-1-13","article-title":"The independence of various definitions of finiteness","volume":"46","author":"L\u00e9vy A.","year":"1958","journal-title":"Fundam. Math."},{"issue":"3","key":"e_1_2_3_8_1","doi-asserted-by":"crossref","first-page":"187","DOI":"10.4064\/fm-84-3-187-208","article-title":"Classes of Dedekind finite cardinals","volume":"84","author":"Truss J. K.","year":"1974","journal-title":"Fundam. Math."}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201300054","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201300054","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,3]],"date-time":"2023-10-03T14:05:11Z","timestamp":1696341911000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201300054"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,7,12]]},"references-count":7,"journal-issue":{"issue":"4-5","published-print":{"date-parts":[[2016,8]]}},"alternative-id":["10.1002\/malq.201300054"],"URL":"https:\/\/doi.org\/10.1002\/malq.201300054","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,7,12]]}}}