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For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice.<\/jats:p>","DOI":"10.2307\/2687776","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:17:03Z","timestamp":1146986223000},"page":"237-261","source":"Crossref","is-referenced-by-count":18,"title":["Relations Between Some Cardinals in the Absence of the Axiom of Choice"],"prefix":"10.1017","volume":"7","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":[[2014,1,15]]},"reference":[{"key":"S1079898600005837_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF01458215"},{"key":"S1079898600005837_ref011","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S1079898600005837_ref023","volume-title":"Cardinal and ordinal numbers","author":"Sierpi\u0144ski","year":"1958"},{"key":"S1079898600005837_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/059"},{"key":"S1079898600005837_ref027","doi-asserted-by":"crossref","first-page":"147","DOI":"10.4064\/fm-5-1-147-154","article-title":"Sur quelques th\u00e9or\u00e8mes qui \u00e9quivalent \u00e0 l'axiome du choix","volume":"5","author":"Tarski","year":"1924","journal-title":"Fundamenta Mathematicae"},{"key":"S1079898600005837_ref010","volume-title":"Grundz\u00fcge der Mengenlehre","author":"Hausdorff","year":"1914"},{"key":"S1079898600005837_ref024","doi-asserted-by":"publisher","DOI":"10.1007\/BF01898373"},{"key":"S1079898600005837_ref016","volume-title":"Set theory, an introduction to independence proofs","author":"Kunen","year":"1983"},{"key":"S1079898600005837_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-88514-3"},{"key":"S1079898600005837_ref021","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9478-5"},{"key":"S1079898600005837_ref022","first-page":"264","article-title":"On a problem of formal logic","volume":"30","author":"Ramsey","year":"1929","journal-title":"Proceedings of the London Mathematical Society, Ser. 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