{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T07:06:30Z","timestamp":1758265590666},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":1962,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2008,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets (x,y) = {{x}, {x,y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a<jats:italic>recursive<\/jats:italic>definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater length. There are many advantages to the new definition, for it allows for uniform definitions working equally well in a wide range of models for set theories. In ZFC and closely related theories, the rank of an ordered pair of two<jats:italic>infinite sets<\/jats:italic>under the new definition turns out to be equal to the maximum of the ranks of the sets.<\/jats:p>","DOI":"10.2178\/bsl\/1231081372","type":"journal-article","created":{"date-parts":[[2009,1,4]],"date-time":"2009-01-04T15:03:06Z","timestamp":1231081386000},"page":"379-397","source":"Crossref","is-referenced-by-count":12,"title":["Reconsidering Ordered Pairs"],"prefix":"10.1017","volume":"14","author":[{"given":"Dana","family":"Scott","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dominic","family":"McCarty","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S107989860000161X_ref005","first-page":"139","volume-title":"Zeitschrift f\u00fcr mathematische Logik und Grundlagen der Mathematik","author":"Boolos","year":"1970"},{"key":"S107989860000161X_ref010","first-page":"273","volume":"9","author":"Kanamori","year":"2003","journal-title":"The empty set, the singleton, and the ordered pair"},{"key":"S107989860000161X_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/2267107"},{"key":"S107989860000161X_ref001","volume-title":"Non-well-founded sets","author":"Aczel","year":"1988"},{"key":"S107989860000161X_ref013","doi-asserted-by":"publisher","DOI":"10.4064\/fm-2-1-161-171"},{"key":"S107989860000161X_ref002","volume-title":"Advances in modal logic","volume":"2","author":"Baltag","year":"2000"},{"key":"S107989860000161X_ref004","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198568520.001.0001"},{"key":"S107989860000161X_ref011","volume-title":"General topology","author":"Kelley","year":"1955"},{"key":"S107989860000161X_ref015","doi-asserted-by":"crossref","DOI":"10.4159\/9780674042445","volume-title":"Philosophy of logic","author":"Quine","year":"1986"},{"key":"S107989860000161X_ref009","volume-title":"Grundz\u00fcge der Mengenlehre","author":"Hausdorff","year":"1914"},{"key":"S107989860000161X_ref016","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10487"},{"key":"S107989860000161X_ref006","volume-title":"Foundations of set theory","volume":"67","author":"Fraenkel","year":"1973"},{"key":"S107989860000161X_ref014","volume-title":"Selected logic papers","author":"Quine","year":"1966"},{"key":"S107989860000161X_ref012","volume-title":"Set theory: An introduction to independence proofs","volume":"102","author":"Kunen","year":"1980"},{"key":"S107989860000161X_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S107989860000161X_ref017","first-page":"224","volume-title":"From Frege to G\u00f6del: A source book in mathematical logic 1879-1931","author":"Wiener","year":"1967"},{"key":"S107989860000161X_ref007","volume-title":"The consistency of the axiom of choice and of the generalised continuum hypothesis","author":"G\u00f6del","year":"1940"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S107989860000161X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,8]],"date-time":"2024-03-08T12:48:03Z","timestamp":1709902083000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S107989860000161X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,9]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,9]]}},"alternative-id":["S107989860000161X"],"URL":"https:\/\/doi.org\/10.2178\/bsl\/1231081372","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,9]]}}}