{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,22]],"date-time":"2024-06-22T18:40:25Z","timestamp":1719081625139},"reference-count":50,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2017,2,27]],"date-time":"2017-02-27T00:00:00Z","timestamp":1488153600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The paradox that appears under Burali\u2013Forti\u2019s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be\u2013absurdly\u2013an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali\u2013Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell\u2019s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali\u2013Forti\u2019s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic.<\/jats:p>","DOI":"10.1017\/s1755020316000484","type":"journal-article","created":{"date-parts":[[2017,2,27]],"date-time":"2017-02-27T08:12:07Z","timestamp":1488183127000},"page":"682-718","source":"Crossref","is-referenced-by-count":6,"title":["WHAT RUSSELL SHOULD HAVE SAID TO BURALI\u2013FORTI"],"prefix":"10.1017","volume":"10","author":[{"given":"SALVATORE","family":"FLORIO","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GRAHAM","family":"LEACH-KROUSE","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,2,27]]},"reference":[{"key":"S1755020316000484_ref40","doi-asserted-by":"publisher","DOI":"10.1111\/j.1520-8583.2003.00014.x"},{"key":"S1755020316000484_ref10","volume-title":"Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics","author":"Ferreir\u00f3s","year":"2007"},{"key":"S1755020316000484_ref7","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1002\/tht3.186","article-title":"The nuisance principle in infinite settings","volume":"4","author":"Ebels-Duggan","year":"2015","journal-title":"Thought"},{"key":"S1755020316000484_ref47","article-title":"Fragments of Frege\u2019s Grundgesetze and G\u00f6dels constructible universe","author":"Walsh","year":"in press","journal-title":"Journal of Symbolic Logic"},{"key":"S1755020316000484_ref29","doi-asserted-by":"publisher","DOI":"10.5840\/jphil2010107311"},{"key":"S1755020316000484_ref19","volume-title":"A New Branch of Mathematics","author":"Grassmann","year":"1847"},{"key":"S1755020316000484_ref16","first-page":"119","volume-title":"Collected Works, Volume II, Publications 1938\u20131974","volume":"1990","author":"G\u00f6del","year":"1944"},{"key":"S1755020316000484_ref38","doi-asserted-by":"publisher","DOI":"10.2307\/2369948"},{"key":"S1755020316000484_ref32","doi-asserted-by":"publisher","DOI":"10.1016\/0315-0860(81)90070-7"},{"key":"S1755020316000484_ref5","first-page":"878","author":"Cantor","year":"1883"},{"key":"S1755020316000484_ref50","first-page":"1219","volume-title":"Fundamenta Mathematicae","volume":"2","author":"Zermelo","year":"1930"},{"key":"S1755020316000484_ref14","doi-asserted-by":"publisher","DOI":"10.1111\/j.1933-1592.2004.tb00518.x"},{"key":"S1755020316000484_ref6","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1082637805"},{"key":"S1755020316000484_ref13","volume-title":"Posthumous Writings","author":"Frege","year":"1979"},{"key":"S1755020316000484_ref39","doi-asserted-by":"publisher","DOI":"10.1093\/mind\/XIX.1.373"},{"key":"S1755020316000484_ref27","volume-title":"Philosophical Papers and Letters","author":"Leibniz","year":"1989"},{"key":"S1755020316000484_ref43","first-page":"253","volume-title":"Absolute Generality","author":"Shapiro","year":"2006"},{"key":"S1755020316000484_ref31","doi-asserted-by":"publisher","DOI":"10.1080\/01445340.2014.967950"},{"key":"S1755020316000484_ref26","article-title":"A generic Russellian elimination of abstract objects","author":"Klement","year":"in press","journal-title":"Philosophia Mathematica"},{"key":"S1755020316000484_ref45","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/axu035"},{"key":"S1755020316000484_ref15","first-page":"20","volume-title":"Absolute Generality","author":"Glanzberg","year":"2006"},{"key":"S1755020316000484_ref44","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"S1755020316000484_ref37","first-page":"29","article-title":"On some difficulties in the theory of transfinite numbers and order types","volume":"4","author":"Russell","year":"1905","journal-title":"Proceedings of the London Mathematical Society"},{"key":"S1755020316000484_ref23","doi-asserted-by":"publisher","DOI":"10.1093\/analys\/anr091"},{"key":"S1755020316000484_ref4","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1878.84.242"},{"key":"S1755020316000484_ref1","first-page":"30","volume-title":"Logic, Logic, and Logic","author":"Boolos","year":"1998"},{"key":"S1755020316000484_ref18","volume-title":"A New Branch of Mathematics. 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