{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T11:20:31Z","timestamp":1772450431859,"version":"3.50.1"},"reference-count":35,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T00:00:00Z","timestamp":1630454400000},"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":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study of certain principles of self-referential truth.<\/jats:p>","DOI":"10.1017\/s1755020321000393","type":"journal-article","created":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T08:50:05Z","timestamp":1630486205000},"page":"385-424","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":5,"title":["SELF-REFERENCE UPFRONT: A STUDY OF SELF-REFERENTIAL G\u00d6DEL NUMBERINGS"],"prefix":"10.1017","volume":"16","author":[{"given":"BALTHASAR","family":"GRABMAYR","sequence":"first","affiliation":[]},{"given":"ALBERT","family":"VISSER","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,9,1]]},"reference":[{"key":"S1755020321000393_r28","volume-title":"Degrees of Unsolvability","author":"Shoenfield","year":"1972"},{"key":"S1755020321000393_r22","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-019-09524-w"},{"key":"S1755020321000393_r4","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-1-35-92"},{"key":"S1755020321000393_r11","doi-asserted-by":"publisher","DOI":"10.1017\/S175502031400029X"},{"key":"S1755020321000393_r9","unstructured":"[9] Halbach, V. , (2018). Personal communication, March 20, 2018."},{"key":"S1755020321000393_r7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22156-3"},{"key":"S1755020321000393_r33","first-page":"149","volume-title":"Handbook of Philosophical Logic","author":"Visser","year":"2004"},{"key":"S1755020321000393_r15","doi-asserted-by":"publisher","DOI":"10.2307\/2024634"},{"key":"S1755020321000393_r19","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/nkm015"},{"key":"S1755020321000393_r10","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020314000288"},{"key":"S1755020321000393_r13","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/nkl028"},{"key":"S1755020321000393_r23","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-019-09525-9"},{"key":"S1755020321000393_r12","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-005-3719-6"},{"key":"S1755020321000393_r14","doi-asserted-by":"publisher","DOI":"10.1007\/s11023-004-6590-1"},{"key":"S1755020321000393_r8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139696586"},{"key":"S1755020321000393_r6","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020320000192"},{"key":"S1755020321000393_r5","doi-asserted-by":"publisher","DOI":"10.2307\/2274093"},{"key":"S1755020321000393_r3","volume-title":"Handbook of Proof Theory","author":"Buss","year":"1998"},{"key":"S1755020321000393_r32","first-page":"617","volume-title":"Handbook of Philosophical Logic, Topics in the Philosophy of Language","author":"Visser","year":"1989"},{"key":"S1755020321000393_r16","unstructured":"[16] Kripke, S. A. , (2021). G\u00f6del\u2019s theorem and direct self-reference. Preprint, 2021, arXiv:2010.11979."},{"key":"S1755020321000393_r18","volume-title":"Parts of Classes","author":"Lewis","year":"1991"},{"key":"S1755020321000393_r17","first-page":"3","article-title":"What is a self-referential sentence?","volume":"177","author":"Leitgeb","year":"2002","journal-title":"Logique et Analyse"},{"key":"S1755020321000393_r29","doi-asserted-by":"publisher","DOI":"10.1007\/BF00485461"},{"key":"S1755020321000393_r34","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2011.01.025"},{"key":"S1755020321000393_r31","doi-asserted-by":"publisher","DOI":"10.1007\/BF00484809"},{"key":"S1755020321000393_r1","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1040609300"},{"key":"S1755020321000393_r30","volume-title":"Undecidable Theories","author":"Tarski","year":"1953"},{"key":"S1755020321000393_r35","unstructured":"[35] Zdanowski, K. (2020). Is there any reasonable non-regular G\u00f6del numbering of the language of arithmetic? MathOverflow. Available from: https:\/\/mathoverflow.net\/q\/361002 (version: 2020-05-22)."},{"key":"S1755020321000393_r24","doi-asserted-by":"publisher","DOI":"10.2307\/2274231"},{"key":"S1755020321000393_r27","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093883561"},{"key":"S1755020321000393_r20","doi-asserted-by":"publisher","DOI":"10.1515\/9781400858927"},{"key":"S1755020321000393_r25","unstructured":"[25] Schindler, T . (2015). Type-Free Truth. 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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"}]}}