{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T20:59:14Z","timestamp":1701205154011},"reference-count":10,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2018,11,19]],"date-time":"2018-11-19T00:00:00Z","timestamp":1542585600000},"content-version":"vor","delay-in-days":0,"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":[[2018,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We write <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0001\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0002\" \/> for the cardinalities of the set of finite sequences and the set of finite subsets, respectively, of a set which is of cardinality <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0003\" \/>. With the axiom of choice (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0004\" \/>), <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0005\" \/> for every infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0006.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0006\" \/> but, without <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0007.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0007\" \/>, any relationship between <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0008.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0008\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0009.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0009\" \/> for an arbitrary infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0010.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0010\" \/> cannot be proved. In this paper, we give conditions that make <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0011.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0011\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0012.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0012\" \/> comparable for an infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0013.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0013\" \/>. Among our results, we show that, if we assume the axiom of choice for sets of finite sets, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0014.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0014\" \/> for every Dedekind\u2010infinite cardinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0015.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0015\" \/> and the condition that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800029-math-0016.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800029:malq201800029-math-0016\" \/> is Dedekind\u2010infinite cannot be weakened to weakly Dedekind\u2010infinite.<\/jats:p>","DOI":"10.1002\/malq.201800029","type":"journal-article","created":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T07:28:07Z","timestamp":1542698887000},"page":"529-534","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Relations between cardinalities of the finite sequences and the finite subsets of a set"],"prefix":"10.1002","volume":"64","author":[{"given":"Navin","family":"Aksornthong","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":[[2018,11,19]]},"reference":[{"key":"e_1_2_5_2_1","first-page":"80","volume-title":"Combinatorial Set Theory With a Gentle Introduction to Forcing","author":"Halbeisen L. 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