{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T17:35:51Z","timestamp":1770399351057,"version":"3.49.0"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2024,12,13]],"date-time":"2024-12-13T00:00:00Z","timestamp":1734048000000},"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":["J. symb. log."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman\u2013Stanley embedding on Scott sentence complexity and show that it only preserves\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481224000598_inline1.png\"\/>\n                        <jats:tex-math>$\\Pi ^{\\mathrm {in}}_{\\alpha }$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    complexities. We then take a more direct approach and exhibit linear orderings of all Scott sentence complexities except\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481224000598_inline2.png\"\/>\n                        <jats:tex-math>$\\Sigma ^{\\mathrm {in}}_{3}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481224000598_inline3.png\"\/>\n                        <jats:tex-math>$\\Sigma ^{\\mathrm {in}}_{\\lambda +1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481224000598_inline4.png\"\/>\n                        <jats:tex-math>$\\lambda $<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    a limit ordinal. We show that the former cannot be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.\n                  <\/jats:p>","DOI":"10.1017\/jsl.2024.59","type":"journal-article","created":{"date-parts":[[2024,12,13]],"date-time":"2024-12-13T06:20:41Z","timestamp":1734070841000},"page":"1563-1592","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["SCOTT SENTENCE COMPLEXITIES OF LINEAR ORDERINGS"],"prefix":"10.1017","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0085-9280","authenticated-orcid":false,"given":"DAVID","family":"GONZALEZ","sequence":"first","affiliation":[{"name":"UNIVERSITY OF CALIFORNIA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3494-9049","authenticated-orcid":false,"given":"DINO","family":"ROSSEGGER","sequence":"additional","affiliation":[{"name":"INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY TECHNISCHE UNIVERSIT\u00c4T WIEN"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2024,12,13]]},"reference":[{"key":"S0022481224000598_r3","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2020.46"},{"key":"S0022481224000598_r22","doi-asserted-by":"publisher","DOI":"10.1017\/9781108525749"},{"key":"S0022481224000598_r5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0860377-7"},{"key":"S0022481224000598_r23","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1981-0624937-1"},{"key":"S0022481224000598_r4","unstructured":"[4] Ash, C. and Knight, J. , Computable Structures and the Hyperarithmetical Hierarchy , Stud. Logic Found. Math. , vol. 144. North-Holland Publishing Co., Amsterdam, 2000."},{"key":"S0022481224000598_r16","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2019.92"},{"key":"S0022481224000598_r17","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2019.91"},{"key":"S0022481224000598_r19","doi-asserted-by":"publisher","DOI":"10.1090\/proc\/12669"},{"key":"S0022481224000598_r12","doi-asserted-by":"publisher","DOI":"10.1201\/9781584887942"},{"key":"S0022481224000598_r1","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2021.4"},{"key":"S0022481224000598_r18","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(02)00035-0"},{"key":"S0022481224000598_r21","doi-asserted-by":"publisher","DOI":"10.1017\/9781108525749"},{"key":"S0022481224000598_r20","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2014.55"},{"key":"S0022481224000598_r14","doi-asserted-by":"crossref","unstructured":"[14] Gonzalez, D. and Montalb\u00e1n, A. , The $\\omega$ -vaught\u2019s conjecture. Trans. Amer. Math. Soc. , vol. 376 (2023), no. 8, pp. 5989\u20136008.","DOI":"10.1090\/tran\/8950"},{"key":"S0022481224000598_r15","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0244049-7"},{"key":"S0022481224000598_r9","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exq041"},{"key":"S0022481224000598_r13","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2005.02.001"},{"key":"S0022481224000598_r11","doi-asserted-by":"publisher","DOI":"10.2307\/2695052"},{"key":"S0022481224000598_r7","doi-asserted-by":"crossref","unstructured":"[7] Calvert, W. , Goncharov, S. S. , and Knight, J. F. , Computable structures of Scott rank ${\\omega}_1^{CK}$ in familiar classes . Advances in logic , vol. 425 (2007), pp. 49\u201366.","DOI":"10.1090\/conm\/425\/08117"},{"key":"S0022481224000598_r8","doi-asserted-by":"publisher","DOI":"10.2307\/2274750"},{"key":"S0022481224000598_r10","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-015-9362-5"},{"key":"S0022481224000598_r2","doi-asserted-by":"publisher","DOI":"10.4064\/fm865-6-2020"},{"key":"S0022481224000598_r6","doi-asserted-by":"publisher","DOI":"10.2307\/2274709"},{"key":"S0022481224000598_r24","doi-asserted-by":"publisher","DOI":"10.2307\/2273222"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481224000598","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T06:57:57Z","timestamp":1770361077000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481224000598\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,13]]},"references-count":24,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["S0022481224000598"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2024.59","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,13]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. 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"}}]}}