{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T08:36:37Z","timestamp":1774946197187,"version":"3.50.1"},"reference-count":78,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T00:00:00Z","timestamp":1756166400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Bull. symb. log"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This article offers a philosophical overview and investigation of the problem of incompleteness in set theory and what this entails for the ensuing debates about proposed extensions of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100930_inline1.png\"\/>\n                        <jats:tex-math>$ZFC$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The incompleteness of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100930_inline2.png\"\/>\n                        <jats:tex-math>$ZFC$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is well-known and leaves us with a rich array of competing extensions. What should we make of disagreements between them? We start by considering second-order logic and its categoricity theorems and how they might be used to compare different set theories. We then aim to use interpretability as a way of understanding that some of these debates are insubstantial. This culminates in some discussion of the relationship between interpretability and the generic multiverse. The second half of the article then takes up a more modest goal: we search for common ground and settle for partial agreement between set theories in much the same way that physicists are often content with empirical agreement. We then aim to describe a natural bound on the amount of agreement that we can expect to obtain between reasonable extensions of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1079898625100930_inline3.png\"\/>\n                        <jats:tex-math>$ZFC$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1017\/bsl.2025.10093","type":"journal-article","created":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T11:51:51Z","timestamp":1756209111000},"page":"619-685","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["OF SHEEP AND WOLVES: EQUIVALENCE AND DISAGREEMENT IN SET THEORY"],"prefix":"10.1017","volume":"31","author":[{"given":"TOBY","family":"MEADOWS","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04gyf1771","id-type":"ROR","asserted-by":"publisher"}],"name":"UNIVERSITY OF CALIFORNIA"}]}],"member":"56","published-online":{"date-parts":[[2025,8,26]]},"reference":[{"key":"S1079898625100930_r23","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020311000359"},{"key":"S1079898625100930_r61","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00021-T"},{"key":"S1079898625100930_r74","volume-title":"100 Years of Russell\u2019s Paradox","author":"Hugh Woodin","year":"2004"},{"key":"S1079898625100930_r75","volume-title":"Infinity: New Research Frontiers","author":"Hugh Woodin","year":"2011"},{"key":"S1079898625100930_r59","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59971-2"},{"key":"S1079898625100930_r53","first-page":"13","article-title":"The uniqueness of the natural numbers","volume":"39","author":"Parsons","year":"1990","journal-title":"Iyyun: The Jerusalem Philosophical Quarterly"},{"key":"S1079898625100930_r38","volume-title":"Naturalism in Mathematics","author":"Maddy","year":"1997"},{"key":"S1079898625100930_r60","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0379200"},{"key":"S1079898625100930_r49","volume-title":"Descriptive Set Theory","author":"Moschovakis","year":"1980"},{"key":"S1079898625100930_r72","first-page":"1","article-title":"When bi-interpretability implies synonymy","volume":"320","author":"Visser","year":"2014","journal-title":"Logic Group Preprint Series"},{"key":"S1079898625100930_r58","doi-asserted-by":"publisher","DOI":"10.2307\/2266389"},{"key":"S1079898625100930_r50","volume-title":"Problems in the Philosophy of Mathematics","author":"Mostowski","year":"1967"},{"key":"S1079898625100930_r69","doi-asserted-by":"publisher","DOI":"10.1142\/S021906131750009X"},{"key":"S1079898625100930_r26","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-2835065"},{"key":"S1079898625100930_r21","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1979-0539907-7"},{"key":"S1079898625100930_r4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01089731"},{"key":"S1079898625100930_r16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9754-0_15"},{"key":"S1079898625100930_r2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-62935-3"},{"key":"S1079898625100930_r54","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1203350787"},{"key":"S1079898625100930_r73","first-page":"567","article-title":"The continuum hypothesis, part I","volume":"48","author":"Woodin","year":"2001","journal-title":"Notices of the AMS"},{"key":"S1079898625100930_r7","volume-title":"Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena","author":"Barwise","year":"1996"},{"key":"S1079898625100930_r3","first-page":"1","author":"Bagaria","year":"2006"},{"key":"S1079898625100930_r31","volume-title":"Set Theory: An Introduction to Independence Proofs","author":"Kunen","year":"2006"},{"key":"S1079898625100930_r48","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-023-04374-1"},{"key":"S1079898625100930_r5","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/axy017"},{"key":"S1079898625100930_r1","unstructured":"[1] Aczel, P. , Non-Well-Founded Sets , CSLI Lecture Notes, CSLI Publications, Palo Alto, 1988."},{"key":"S1079898625100930_r28","volume-title":"The Axiom of Choice","author":"Jech","year":"2008"},{"key":"S1079898625100930_r36","first-page":"329","volume-title":"Proceedings of the 5th Scandinavian Logic Symposium","author":"Lindstr\u00f6m","year":"1979"},{"key":"S1079898625100930_r8","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198790396.001.0001"},{"key":"S1079898625100930_r66","doi-asserted-by":"publisher","DOI":"10.1017\/9781108886840"},{"key":"S1079898625100930_r6","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020316000186"},{"key":"S1079898625100930_r63","unstructured":"[63] Steel, J. 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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"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}