{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T13:49:10Z","timestamp":1773150550375,"version":"3.50.1"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7316,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline1\"\/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline2\"\/>, either <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline1\"\/> \u2264 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline2\"\/> or <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline2\"\/> \u2264 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline1\"\/>. However, in ZF this is no longer so. For a given infinite set <jats:italic>A<\/jats:italic> consider seq<jats:sup>1-1<\/jats:sup>(<jats:italic>A<\/jats:italic>), the set of all sequences of <jats:italic>A<\/jats:italic> without repetition. We compare |seq<jats:sup>1-1<\/jats:sup>(<jats:italic>A<\/jats:italic>)|, the cardinality of this set, to |<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline3\"\/>|, the cardinality of the power set of <jats:italic>A<\/jats:italic>. What is provable about these two cardinals in ZF? The main result of this paper is that ZF \u22a2 \u2200<jats:italic>A<\/jats:italic>(| seq<jats:sup>1-1<\/jats:sup>(<jats:italic>A<\/jats:italic>)| \u2260 |<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline3\"\/>|), and we show that this is the best possible result. Furthermore, it is provable in ZF that if <jats:italic>B<\/jats:italic> is an infinite set, then | fin(B)| &lt; | <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline4\"\/><jats:italic>(B*)<\/jats:italic>| even though the existence for some infinite set <jats:italic>B<\/jats:italic>* of a function <jats:italic>\u0192<\/jats:italic> from fin(<jats:italic>B<\/jats:italic>*) onto <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002020X_inline4\"\/>(<jats:italic>B<\/jats:italic>*) is consistent with ZF.<\/jats:p>","DOI":"10.2307\/2275247","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:51:08Z","timestamp":1146955868000},"page":"30-40","source":"Crossref","is-referenced-by-count":27,"title":["Consequences of arithmetic for set theory"],"prefix":"10.1017","volume":"59","author":[{"given":"Lorenz","family":"Halbeisen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saharon","family":"Shelah","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120002020X_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01898373"},{"key":"S002248120002020X_ref003","volume-title":"The axiom of choice","author":"Jech","year":"1973"},{"key":"S002248120002020X_ref002","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S002248120002020X_ref005","volume-title":"A handbook of integer sequence","author":"Sloane","year":"1973"},{"key":"S002248120002020X_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-88514-3"},{"key":"S002248120002020X_ref004","first-page":"1\u201318","article-title":"Auswahlaxiom in der Algebra","volume":"37","author":"L\u00e4uchli","year":"1962","journal-title":"Commentarii Mathematici Hehetiei"},{"key":"S002248120002020X_ref007","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19570031302"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120002020X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T19:16:39Z","timestamp":1557947799000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120002020X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1994,3]]}},"alternative-id":["S002248120002020X"],"URL":"https:\/\/doi.org\/10.2307\/2275247","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,3]]}}}