{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T00:15:33Z","timestamp":1724890533412},"reference-count":38,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T00:00:00Z","timestamp":1618876800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2021,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The<jats:italic>injective<\/jats:italic>version of Cantor\u2019s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege\u2019s Basic Law V (<jats:monospace>BLV<\/jats:monospace>), an inconsistency easily shown using Russell\u2019s paradox. This incompatibility is akin to others\u2014most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali\u2013Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice\u2014a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali\u2013Fortian method of proof.<\/jats:p>","DOI":"10.1017\/s1755020318000230","type":"journal-article","created":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T09:28:07Z","timestamp":1618910887000},"page":"592-623","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["DEDUCTIVE CARDINALITY RESULTS AND NUISANCE-LIKE PRINCIPLES"],"prefix":"10.1017","volume":"14","author":[{"given":"SEAN C.","family":"EBELS-DUGGAN","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,4,20]]},"reference":[{"key":"S1755020318000230_r30","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1038336880"},{"key":"S1755020318000230_r24","first-page":"331","article-title":"Some criteria for acceptable abstraction","volume":"52","author":"Linnebo","year":"2010","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020318000230_r22","doi-asserted-by":"publisher","DOI":"10.2307\/421146"},{"volume-title":"Essays in Analysis","year":"1973","author":"Russell","key":"S1755020318000230_r28"},{"key":"S1755020318000230_r23","first-page":"413","volume-title":"Paul Erd\u00f6s and his mathematics, volume\u00a02 of Bolyai Society Mathematical Studies","author":"Kanamori","year":"2002"},{"key":"S1755020318000230_r36","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1093\/oso\/9780198239208.003.0008","volume-title":"Language, Thought, and Logic: Essays in Honour of Michael Dummett","author":"Wright","year":"1997"},{"key":"S1755020318000230_r15","doi-asserted-by":"publisher","DOI":"10.1525\/9780520312364"},{"key":"S1755020318000230_r31","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/7.3.293"},{"key":"S1755020318000230_r12","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: A Journal of Philosophy"},{"volume-title":"Basic Laws of Arithmetic","year":"2013","author":"Frege","key":"S1755020318000230_r17"},{"key":"S1755020318000230_r2","first-page":"1","article-title":"Purity of methods","volume":"11","author":"Arana","year":"2011","journal-title":"Philosopher\u2019s Imprint"},{"key":"S1755020318000230_r19","doi-asserted-by":"publisher","DOI":"10.1093\/0198236395.001.0001"},{"volume-title":"The Frege Reader","year":"1997","author":"Beaney","key":"S1755020318000230_r3"},{"key":"S1755020318000230_r9","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/axr047"},{"key":"S1755020318000230_r7","volume-title":"The Western Ontario Series in Philosophy of Science.","volume":"71","author":"Cook","year":"2007"},{"key":"S1755020318000230_r26","first-page":"29","article-title":"On some difficulties in the theory of transfinite numbers and order types","volume":"4","author":"Russell","year":"1906","journal-title":"Proceedings of the London Mathematical Society"},{"key":"S1755020318000230_r25","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199252619.001.0001"},{"key":"S1755020318000230_r33","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50014-8"},{"volume-title":"The Principles of Mathematics","year":"1913","author":"Russell","key":"S1755020318000230_r27"},{"key":"S1755020318000230_r37","first-page":"6","article-title":"Is Hume\u2019s principle analytic?","volume":"40","author":"Wright","year":"1998","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020318000230_r21","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004299720847"},{"volume-title":"The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number","year":"1980","author":"Frege","key":"S1755020318000230_r16"},{"key":"S1755020318000230_r4","doi-asserted-by":"publisher","DOI":"10.5840\/philtopics19891721"},{"key":"S1755020318000230_r5","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004209106100"},{"key":"S1755020318000230_r1","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/nkq010"},{"key":"S1755020318000230_r8","first-page":"273","volume-title":"Reduction, Abstraction, Analysis, Proceedings of the 31 th International Ludwig Wittgentstein Symposium in Kirchberg, 2008","author":"Cook","year":"2009"},{"volume-title":"Essays on the Theory of Numbers","year":"1963","author":"Dedekind","key":"S1755020318000230_r11"},{"key":"S1755020318000230_r10","doi-asserted-by":"publisher","DOI":"10.1007\/s001530100074"},{"key":"S1755020318000230_r13","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-2018-0023"},{"key":"S1755020318000230_r38","first-page":"139","volume-title":"From Frege to G\u00f6del: A Source Book in Mathematical Logic","author":"Zermelo","year":"1904"},{"key":"S1755020318000230_r6","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1093\/oso\/9780198239208.003.0009","volume-title":"Language, Thought, and Logic: Essays in Honour of Michael Dummett","author":"Boolos","year":"1997"},{"key":"S1755020318000230_r18","doi-asserted-by":"publisher","DOI":"10.1016\/0315-0860(78)90046-0"},{"key":"S1755020318000230_r34","doi-asserted-by":"publisher","DOI":"10.1142\/8962"},{"key":"S1755020318000230_r14","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780199246182.001.0001","volume-title":"The Limits of Abstraction","author":"Fine","year":"2002"},{"key":"S1755020318000230_r35","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1082637613"},{"volume-title":"A Source Book in Mathematical Logic","year":"1967","author":"van Heijenoort","key":"S1755020318000230_r32"},{"volume-title":"Foundations without Foundationalism: A Case for Second-Order Logic, volume\u00a017 of Oxford Logic Guides","year":"1991","author":"Shapiro","key":"S1755020318000230_r29"},{"key":"S1755020318000230_r20","doi-asserted-by":"publisher","DOI":"10.2307\/2216025"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020318000230","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T20:40:00Z","timestamp":1724877600000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020318000230\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,20]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["S1755020318000230"],"URL":"https:\/\/doi.org\/10.1017\/s1755020318000230","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"type":"print","value":"1755-0203"},{"type":"electronic","value":"1755-0211"}],"subject":[],"published":{"date-parts":[[2021,4,20]]},"assertion":[{"value":"\u00a9 Association for Symbolic Logic, 2021","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}