{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T13:03:03Z","timestamp":1770814983423,"version":"3.50.1"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2021,3,22]],"date-time":"2021-03-22T00:00:00Z","timestamp":1616371200000},"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":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000095_inline1.png\"\/><jats:tex-math>\n$\\mathsf {CH}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse.<\/jats:p>","DOI":"10.1017\/s1755020321000095","type":"journal-article","created":{"date-parts":[[2021,3,22]],"date-time":"2021-03-22T07:18:13Z","timestamp":1616397493000},"page":"979-1010","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":6,"title":["INDEPENDENCE PROOFS IN NON-CLASSICAL SET THEORIES"],"prefix":"10.1017","volume":"16","author":[{"given":"SOURAV","family":"TARAFDER","sequence":"first","affiliation":[]},{"given":"GIORGIO","family":"VENTURI","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,3,22]]},"reference":[{"key":"S1755020321000095_r4","doi-asserted-by":"publisher","DOI":"10.2307\/2025204"},{"key":"S1755020321000095_r2","doi-asserted-by":"publisher","DOI":"10.1007\/s40961-017-0102-1"},{"key":"S1755020321000095_r6","first-page":"402","volume-title":"Applications of Sheaves, Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis Held at the University of Durham, Durham, July 9\u201321, 1977.","volume":"753","author":"Grayson","year":"1979"},{"key":"S1755020321000095_r8","doi-asserted-by":"publisher","DOI":"10.1017\/9781108596961"},{"key":"S1755020321000095_r17","doi-asserted-by":"crossref","unstructured":"17 Tarafder, S. (2021). Non-classical foundations of set theory, Submitted.","DOI":"10.1017\/jsl.2021.101"},{"key":"S1755020321000095_r21","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511976889.006"},{"key":"S1755020321000095_r1","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-015-0819-9"},{"key":"S1755020321000095_r7","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020311000359"},{"key":"S1755020321000095_r16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-45824-2_14"},{"key":"S1755020321000095_r10","article-title":"Non-classical models of $\\mathsf{ZF}$","author":"Jockwich","year":"2020a","journal-title":"Studia Logica"},{"key":"S1755020321000095_r14","volume-title":"Foundations of Mathematics","author":"Maddy","year":"2016"},{"key":"S1755020321000095_r18","doi-asserted-by":"publisher","DOI":"10.1007\/978-81-322-2719-9_7"},{"key":"S1755020321000095_r3","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198568520.001.0001"},{"key":"S1755020321000095_r19","doi-asserted-by":"publisher","DOI":"10.1111\/j.1758-2237.2002.tb00067.x"},{"key":"S1755020321000095_r20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-31644-4_12"},{"key":"S1755020321000095_r13","doi-asserted-by":"publisher","DOI":"10.1017\/S175502031400046X"},{"key":"S1755020321000095_r5","doi-asserted-by":"publisher","DOI":"10.2307\/420965"},{"key":"S1755020321000095_r12","unstructured":"12 L\u00f6we, B. , Pa\u00dfmann, R. , & Tarafder, S. (2018). Constructing illoyal algebra-valued models of set theory. Algebra Universalis, accepted."},{"key":"S1755020321000095_r11","article-title":"On negation for non-classical set theories","author":"Jockwich","year":"2020b","journal-title":"Journal of Philosophical Logic"},{"key":"S1755020321000095_r15","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-016-1297-4"},{"key":"S1755020321000095_r9","unstructured":"9 Jockwich, S. , Tarafder, S. , & Venturi, G. (2020). $\\mathsf{ZF}$ and its interpretations. Submitted."}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020321000095","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T00:00:08Z","timestamp":1703030408000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020321000095\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,22]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["S1755020321000095"],"URL":"https:\/\/doi.org\/10.1017\/s1755020321000095","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,22]]},"assertion":[{"value":"\u00a9 The Author(s), 2021. 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"}}]}}