{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T22:04:29Z","timestamp":1747173869426,"version":"3.40.5"},"reference-count":47,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T00:00:00Z","timestamp":1682380800000},"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":[[2024,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000102_inline1.png\"\/><jats:tex-math>\n$H_{\\kappa ^+}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000102_inline2.png\"\/><jats:tex-math>\n$\\kappa $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> ranges among the infinite cardinals. We also single out <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000102_inline3.png\"\/><jats:tex-math>\n$2^{\\aleph _0}=\\aleph _2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000102_inline4.png\"\/><jats:tex-math>\n$H_{\\aleph _2}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> as given by a strong form of Woodin\u2019s axiom <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000102_inline5.png\"\/><jats:tex-math>\n$(*)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (which holds assuming <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000102_inline6.png\"\/><jats:tex-math>\n$\\mathsf {MM}^{++}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert\u2019s axiom of completeness.<\/jats:p>","DOI":"10.1017\/s1755020323000102","type":"journal-article","created":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T10:34:59Z","timestamp":1682418899000},"page":"546-585","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["WHAT MODEL COMPANIONSHIP CAN SAY ABOUT THE CONTINUUM PROBLEM"],"prefix":"10.1017","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9089-4340","authenticated-orcid":false,"given":"GIORGIO","family":"VENTURI","sequence":"first","affiliation":[]},{"given":"MATTEO","family":"VIALE","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,4,25]]},"reference":[{"key":"S1755020323000102_r13","volume-title":"Model Theory","volume":"73","author":"Chen Chang","year":"1990"},{"key":"S1755020323000102_r34","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139015417"},{"key":"S1755020323000102_r6","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050154"},{"key":"S1755020323000102_r32","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061305000407"},{"key":"S1755020323000102_r28","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1111\/nous.12208","article-title":"Actual and potential infinity","volume":"53","author":"Shapiro","year":"2019","journal-title":"No\u00fbs"},{"key":"S1755020323000102_r35","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2002.v9.n4.a6"},{"key":"S1755020323000102_r36","first-page":"435","article-title":"Genericity and arbitrariness","volume":"248","author":"Venturi","year":"2019","journal-title":"Logique et Analyse"},{"key":"S1755020323000102_r30","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199273669.001.0001"},{"key":"S1755020323000102_r39","doi-asserted-by":"publisher","DOI":"10.1090\/jams\/844"},{"key":"S1755020323000102_r40","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-015-0466-3"},{"volume-title":"Naturalism in Mathematics","year":"1997","author":"Maddy","key":"S1755020323000102_r29"},{"key":"S1755020323000102_r18","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1052669062"},{"volume-title":"S\u00e9minaire Bourbaki","year":"2023","author":"Viale","key":"S1755020323000102_r42"},{"key":"S1755020323000102_r45","first-page":"567","article-title":"The continuum hypothesis. 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