{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T07:51:22Z","timestamp":1773820282886,"version":"3.50.1"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":284,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2013,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For any ordinal \u039b, we can define a polymodal logic GLP<jats:sub>\u039b<\/jats:sub>, with a modality [<jats:italic>\u03be<\/jats:italic>] for each <jats:italic>\u03be<\/jats:italic> &lt; \u039b. These represent provability predicates of increasing strength. Although GLP<jats:sub>\u039b<\/jats:sub> has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000013X_inline1\"\/>. Later, Icard defined a topological model for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000013X_inline1\"\/> which is very closely related to Ignatiev's.<\/jats:p><jats:p>In this paper we show how to extend these constructions for arbitrary \u039b. More generally, for each \u0398, \u039b we build a Kripke model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000013X_inline3\"\/> and a topological model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000013X_inline4\"\/>, and show that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000013X_inline5\"\/> is sound for both of these structures, as well as complete, provided \u0398 is large enough.<\/jats:p>","DOI":"10.2178\/jsl.7802110","type":"journal-article","created":{"date-parts":[[2013,5,15]],"date-time":"2013-05-15T10:11:08Z","timestamp":1368612668000},"page":"543-561","source":"Crossref","is-referenced-by-count":16,"title":["Models of transfinite provability logic"],"prefix":"10.1017","volume":"78","author":[{"given":"David","family":"Fern\u00e1ndez-Duque","sequence":"first","affiliation":[]},{"given":"Joost J.","family":"Joosten","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000013X_ref004","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exi038"},{"key":"S002248120000013X_ref005","volume-title":"Hyperations, Vehlen progressions and transfinite iteration of ordinal functions","author":"Fern\u00e1ndez-Duque","year":"2012"},{"key":"S002248120000013X_ref011","first-page":"249","volume":"58","author":"Ignatiev","year":"1993","journal-title":"On strong provability predicates and the associated modal logics"},{"key":"S002248120000013X_ref010","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exp043"},{"key":"S002248120000013X_ref012","volume-title":"Intensional logics and logical structure of theories: material from the Fourth Soviet-Finnish symposium on logic","author":"Japaridze","year":"1988"},{"key":"S002248120000013X_ref014","volume-title":"Filosofiska F\u00f6reningen och Filosofiska Institutienen vid Uppsala Universitet","author":"Segerberg","year":"1971"},{"key":"S002248120000013X_ref002","first-page":"1","volume-title":"GLP, ArXiv","author":"Beklemishev","year":"2011"},{"key":"S002248120000013X_ref008","volume-title":"Well-orders in the transfinite Japaridze algebra","author":"Fern\u00e1ndez-Duque","year":"2012"},{"key":"S002248120000013X_ref006","first-page":"185","volume-title":"Advances in modal logic","volume":"9","author":"Fern\u00e1ndez-Duque","year":"2012"},{"key":"S002248120000013X_ref001","volume-title":"On provability logics with linearly ordered modalities","author":"Beklemishev","year":"2012"},{"key":"S002248120000013X_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-30870-3_21"},{"key":"S002248120000013X_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2003.11.030"},{"key":"S002248120000013X_ref009","volume-title":"Well-orders in the transfinite Japaridze algebra II: Turing progressions and their well-orders","author":"Fern\u00e1ndez-Duque","year":"2012"},{"key":"S002248120000013X_ref013","unstructured":"[] Joosten J. J. , lntepretability formalized. Department of Philosophy, University of Utrecht, 2004, Ph.D. thesis."},{"key":"S002248120000013X_ref015","first-page":"33","article-title":"Provability interpretations of modal logic","volume":"28","author":"Solovay","year":"1976","journal-title":"Israel Journal of Mathematics"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120000013X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T17:40:13Z","timestamp":1556041213000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120000013X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,6]]},"references-count":15,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,6]]}},"alternative-id":["S002248120000013X"],"URL":"https:\/\/doi.org\/10.2178\/jsl.7802110","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,6]]}}}