{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T11:10:54Z","timestamp":1684235454309},"reference-count":35,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T00:00:00Z","timestamp":1598313600000},"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":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume\u2019s Principle is analytic then in the standard setting the answer appears to be yes. Hodes\u2019s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.<\/jats:p>","DOI":"10.1017\/s1755020320000349","type":"journal-article","created":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T03:40:54Z","timestamp":1598326854000},"page":"553-577","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["THE POTENTIAL IN FREGE\u2019S THEOREM"],"prefix":"10.1017","volume":"16","author":[{"given":"WILL","family":"STAFFORD","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,8,25]]},"reference":[{"key":"S1755020320000349_r6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-4265-2"},{"key":"S1755020320000349_r9","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-5292-1"},{"key":"S1755020320000349_r16","doi-asserted-by":"publisher","DOI":"10.1007\/BF00485187"},{"key":"S1755020320000349_r28","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020315000039"},{"key":"S1755020320000349_r22","doi-asserted-by":"publisher","DOI":"10.1111\/nous.12208"},{"key":"S1755020320000349_r33","doi-asserted-by":"publisher","DOI":"10.1017\/bsl.2015.39"},{"key":"S1755020320000349_r34","volume-title":"Principia Mathematica","volume":"1","author":"Whitehead","year":"1910"},{"key":"S1755020320000349_r35","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199552078.001.0001"},{"key":"S1755020320000349_r27","doi-asserted-by":"publisher","DOI":"10.2307\/2024603"},{"key":"S1755020320000349_r1","doi-asserted-by":"publisher","DOI":"10.2307\/2586481"},{"key":"S1755020320000349_r25","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200111)47:4<513::AID-MALQ513>3.0.CO;2-J"},{"key":"S1755020320000349_r32","doi-asserted-by":"publisher","DOI":"10.3390\/axioms5020018"},{"key":"S1755020320000349_r20","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020313000014"},{"key":"S1755020320000349_r24","doi-asserted-by":"publisher","DOI":"10.1111\/nous.12208"},{"key":"S1755020320000349_r19","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-015-9389-z"},{"key":"S1755020320000349_r12","volume-title":"Metamathematics of First-Order Arithmetic","author":"H\u00e1jek","year":"1998"},{"key":"S1755020320000349_r30","first-page":"555","volume-title":"A Companion to the Philosophy of Language","author":"Stanley","year":"1997"},{"key":"S1755020320000349_r23","volume-title":"From Frege to G\u00f6del : A Source Book in Mathematical Logic, 1879\u20131931","author":"van Heijenoort","year":"1967"},{"key":"S1755020320000349_r10","volume-title":"Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung \u00fcber den Begriff der Zahl","author":"Frege","year":"1884"},{"key":"S1755020320000349_r15","doi-asserted-by":"publisher","DOI":"10.2307\/2026440"},{"key":"S1755020320000349_r2","doi-asserted-by":"publisher","DOI":"10.2307\/2183530"},{"key":"S1755020320000349_r31","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/axu035"},{"key":"S1755020320000349_r5","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198790396.001.0001"},{"key":"S1755020320000349_r7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01048481"},{"key":"S1755020320000349_r21","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780199641314.001.0001"},{"key":"S1755020320000349_r17","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S1755020320000349_r13","doi-asserted-by":"publisher","DOI":"10.1093\/0198236395.001.0001"},{"key":"S1755020320000349_r18","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-014-9551-6"},{"key":"S1755020320000349_r11","volume-title":"Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet","author":"Frege","year":"1893"},{"key":"S1755020320000349_r14","doi-asserted-by":"publisher","DOI":"10.2307\/2275220"},{"key":"S1755020320000349_r26","first-page":"298","volume-title":"Mathematics in Philosophy: Selected Essays","author":"Parsons","year":"1983"},{"key":"S1755020320000349_r3","volume-title":"Logic, Logic, and Logic","author":"Boolos","year":"1998"},{"key":"S1755020320000349_r29","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"S1755020320000349_r4","doi-asserted-by":"publisher","DOI":"10.1515\/9780691187068"},{"key":"S1755020320000349_r8","doi-asserted-by":"publisher","DOI":"10.1093\/0195148770.003.0019"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020320000349","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T10:34:00Z","timestamp":1684233240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020320000349\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,25]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["S1755020320000349"],"URL":"https:\/\/doi.org\/10.1017\/s1755020320000349","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,25]]},"assertion":[{"value":"\u00a9 The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}