{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T09:26:56Z","timestamp":1679650016360},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2020,7,10]],"date-time":"2020-07-10T00:00:00Z","timestamp":1594339200000},"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":[[2021,12]]},"abstract":"Abstract<\/jats:title>It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\\Pi ^1_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>reflection strength order. We prove that there are no descending sequences of$\\Pi ^1_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>sound extensions of$\\mathsf {ACA}_0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\\Pi ^1_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>sound extension of$\\mathsf {ACA}_0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We prove that for any$\\Pi ^1_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>sound theoryT<\/jats:italic>extending$\\mathsf {ACA}_0^+$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the reflection rank ofT<\/jats:italic>equals the$\\Pi ^1_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>proof-theoretic ordinal ofT<\/jats:italic>. We also prove that the$\\Pi ^1_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>proof-theoretic ordinal of$\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>iterated$\\Pi ^1_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>reflection is$\\varepsilon _\\alpha $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.<\/jats:p>","DOI":"10.1017\/jsl.2020.9","type":"journal-article","created":{"date-parts":[[2020,7,10]],"date-time":"2020-07-10T09:59:02Z","timestamp":1594375142000},"page":"1350-1384","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":5,"title":["REFLECTION RANKS AND ORDINAL ANALYSIS"],"prefix":"10.1017","volume":"86","author":[{"given":"FEDOR","family":"PAKHOMOV","sequence":"first","affiliation":[]},{"given":"JAMES","family":"WALSH","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,7,10]]},"reference":[{"key":"S0022481220000092_r26","volume-title":"Self-Reference and Modal Logic","author":"Smorynski","year":"2012"},{"key":"S0022481220000092_r8","doi-asserted-by":"publisher","DOI":"10.1070\/RM9843"},{"key":"S0022481220000092_r16","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19680140702"},{"key":"S0022481220000092_r9","unstructured":"[9] Beklemishev, L. and Pakhomov, F. , Reflection algebras and conservation results for theories of iterated truth, 2019. arXiv:1908.10302."},{"key":"S0022481220000092_r6","first-page":"89","article-title":"Calibrating provability logic: From modal logic to reflection calculus","volume":"9","author":"Beklemishev","year":"2012","journal-title":"Advances in Modal Logic"},{"key":"S0022481220000092_r23","first-page":"721","article-title":"Iterated reflection principles and the\u00a0 $\\omega$ -rule","volume":"47","author":"Schmerl","year":"1982","journal-title":"this Journal"},{"key":"S0022481220000092_r21","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325944.011"},{"key":"S0022481220000092_r5","first-page":"65","volume-title":"Logic, Methodology and Philosophy of Science, Proceedings of the Twelfth International Congress","author":"Beklemishev","year":"2005"},{"key":"S0022481220000092_r2","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-002-0158-7"},{"key":"S0022481220000092_r25","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90074-6"},{"key":"S0022481220000092_r3","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2003.11.030"},{"key":"S0022481220000092_r15","doi-asserted-by":"crossref","DOI":"10.1017\/9781316717271","volume-title":"Metamathematics of First-Order Arithmetic","author":"H\u00e1jek","year":"2017"},{"key":"S0022481220000092_r7","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2013.07.006"},{"key":"S0022481220000092_r4","doi-asserted-by":"publisher","DOI":"10.1070\/RM2005v060n02ABEH000823"},{"key":"S0022481220000092_r24","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"S0022481220000092_r10","doi-asserted-by":"publisher","DOI":"10.1134\/S0001434612030029"},{"key":"S0022481220000092_r17","doi-asserted-by":"crossref","DOI":"10.1017\/9781316716854","volume-title":"Aspects of Incompleteness","author":"Lindstr\u00f6m","year":"2017"},{"key":"S0022481220000092_r13","first-page":"363","article-title":"Uniformly defined descending sequences of degrees","volume":"41","author":"Friedman","year":"1976","journal-title":"this Journal"},{"key":"S0022481220000092_r22","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71633-1"},{"key":"S0022481220000092_r27","first-page":"59","article-title":"Descending sequences of degrees","volume":"40","author":"Steel","year":"1975","journal-title":"this Journal"},{"key":"S0022481220000092_r11","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-018-0657-9"},{"key":"S0022481220000092_r18","unstructured":"[18] Lutz, P. and Walsh, J. , Incompleteness and jump hierarchies, 2019. arXiv:1909.10603."},{"key":"S0022481220000092_r20","volume-title":"Proof Theory: The First Step into Impredicativity","author":"Pohlers","year":"2008"},{"key":"S0022481220000092_r12","unstructured":"[12] Fern\u00e1ndez-Duque, D. , Worms and spiders: Reflection calculi and ordinal notation systems, 2016. arXiv:1605.08867."},{"key":"S0022481220000092_r1","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(95)00007-4"},{"key":"S0022481220000092_r14","unstructured":"[14] Frittaion, E. , Uniform reflection in second order arithmetic, 2019. Available at https:\/\/drive.google.com\/file\/d\/19-25_Gr5wGE6beQD5ho_k52slivagTjU\/view."},{"key":"S0022481220000092_r19","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2020.9"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481220000092","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,2]],"date-time":"2022-11-02T06:49:57Z","timestamp":1667371797000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481220000092\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,10]]},"references-count":27,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["S0022481220000092"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2020.9","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,10]]},"assertion":[{"value":"\u00a9 Association for Symbolic Logic 2020","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}