{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T04:45:34Z","timestamp":1772426734121,"version":"3.50.1"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T00:00:00Z","timestamp":1760918400000},"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":["J. symb. log."],"published-print":{"date-parts":[[2026,3]]},"abstract":"Abstract<\/jats:title>\n \n Motivated by ideas from the model theory of metric structures, we introduce a metric set theory,\n \n \n \n $\\mathsf {MSE}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , which takes bounded quantification as primitive and consists of a natural metric extensionality axiom (the distance between two sets is the Hausdorff distance between their extensions) and an approximate, non-deterministic form of full comprehension (for any real-valued formula\n \n \n \n $\\varphi (x,y)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , tuple of parameters\n a<\/jats:italic>\n , and\n \n \n \n $r < s$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , there is a set containing the class\n \n and contained in the class\n \n \n \n $\\{x:\\varphi (x,a) < s\\}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ). We show that\n \n \n \n $\\mathsf {MSE}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is sufficient to develop classical mathematics after the addition of an appropriate axiom of infinity. We then construct canonical representatives of well-order types and prove that ultrametric models of\n \n \n \n $\\mathsf {MSE}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n always contain externally ill-founded ordinals, conjecturing that this is true of all models. To establish several independence results and, in particular, consistency, we construct a variety of models, including pseudo-finite models and models containing arbitrarily large standard ordinals. Finally, we discuss how to formalize\n \n \n \n $\\mathsf {MSE}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in either continuous logic or \u0141ukasiewicz logic.\n <\/jats:p>","DOI":"10.1017\/jsl.2025.10147","type":"journal-article","created":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T10:58:35Z","timestamp":1760957915000},"page":"129-174","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A METRIC SET THEORY WITH A UNIVERSAL SET"],"prefix":"10.1017","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9269-3446","authenticated-orcid":false,"given":"JAMES E.","family":"HANSON","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04rswrd78","id-type":"ROR","asserted-by":"publisher"}],"name":"IOWA STATE UNIVERSITY"}]}],"member":"56","published-online":{"date-parts":[[2025,10,20]]},"reference":[{"key":"S0022481225101473_r15","volume-title":"Set theory in which the axiom of foundation fails","author":"Malitz","year":"1976"},{"key":"S0022481225101473_r18","volume-title":"The Stanford Encyclopedia of Philosophy","author":"Randall Holmes","year":"2021"},{"key":"S0022481225101473_r9","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exab036"},{"key":"S0022481225101473_r3","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10685"},{"key":"S0022481225101473_r24","doi-asserted-by":"publisher","DOI":"10.1007\/BF00258447"},{"key":"S0022481225101473_r7","doi-asserted-by":"publisher","DOI":"10.2307\/2586560"},{"key":"S0022481225101473_r11","volume-title":"Definability and categoricity in continuous logic","author":"Hanson","year":"2020"},{"key":"S0022481225101473_r14","unstructured":"[14] Jerome Keisler, H. , Model theory for real-valued structures, preprint, 2020, arXiv:2005.11851."},{"key":"S0022481225101473_r13","doi-asserted-by":"publisher","DOI":"10.1007\/BF00568059"},{"key":"S0022481225101473_r10","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2022.103204"},{"key":"S0022481225101473_r16","doi-asserted-by":"publisher","DOI":"10.2307\/2268660"},{"key":"S0022481225101473_r17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-06233-4"},{"key":"S0022481225101473_r2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2014.01.005"},{"key":"S0022481225101473_r8","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-005-0284-0"},{"key":"S0022481225101473_r21","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139015417"},{"key":"S0022481225101473_r19","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19570030102"},{"key":"S0022481225101473_r23","volume-title":"How to approximate the naive comprehension scheme inside of classical logic","author":"Weydert","year":"1989"},{"key":"S0022481225101473_r6","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(95)00087-9"},{"key":"S0022481225101473_r22","unstructured":"[22] Terui, K. , A flaw in R.B. White\u2019s article \u201cThe consistency of the axiom of comprehension in the infinite-valued predicate logic of \u0141ukasiewicz\u201d. 2014, Unpublished."},{"key":"S0022481225101473_r1","volume-title":"Beyond First Order Model Theory","volume":"1","author":"Caicedo","year":"2020"},{"key":"S0022481225101473_r20","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.39.9.972"},{"key":"S0022481225101473_r12","unstructured":"[12] Holmes, M. R. and Wilshaw, S. , NF is consistent, preprint, 2025."},{"key":"S0022481225101473_r4","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10706"},{"key":"S0022481225101473_r25","first-page":"315","volume-title":"Model Theory for Metric Structures","volume":"2","author":"Yaacov","year":"2008"},{"key":"S0022481225101473_r5","volume-title":"Set Theory with a Universal Set: Exploring an Untyped Universe","author":"Forster","year":"1992"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481225101473","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T02:28:54Z","timestamp":1772418534000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481225101473\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,20]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0022481225101473"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2025.10147","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,20]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. 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"}]}}