{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T04:48:12Z","timestamp":1681706892344},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4940,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2000,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we introduce a recursive notation system <jats:italic>O<\/jats:italic>(\u03a0<jats:sub>3<\/jats:sub>) of ordinals. An element of the notation system is called an <jats:italic>ordinal diagram<\/jats:italic>. The system is designed for proof theoretic study of theories of \u03a0<jats:sub>3<\/jats:sub>-reflection. We show that for each \u03b1 &lt; \u03a9 in <jats:italic>O<\/jats:italic>(\u03a0<jats:sub>3<\/jats:sub>) a set theory KP \u03a0<jats:sub>3<\/jats:sub> for \u03a0<jats:sub>3<\/jats:sub>-reflection proves that the initial segment of <jats:italic>O<\/jats:italic>(\u03a0<jats:sub>3<\/jats:sub>) determined by \u03b1 is a well ordering. Proof theoretic study for such theories will be reported in [ 4].<\/jats:p>","DOI":"10.2307\/2586705","type":"journal-article","created":{"date-parts":[[2006,11,18]],"date-time":"2006-11-18T09:45:07Z","timestamp":1163843107000},"page":"1375-1394","source":"Crossref","is-referenced-by-count":6,"title":["Ordinal diagrams for \u03a0<sub>3<\/sub>-reflection"],"prefix":"10.1017","volume":"65","author":[{"given":"Toshiyasu","family":"Arai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200011968_ref009","first-page":"301\u2013381","volume-title":"Generalized recursion theory","volume":"79","author":"Richter","year":"1974"},{"key":"S0022481200011968_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)90074-4"},{"key":"S0022481200011968_ref005","first-page":"4\u201325","volume-title":"Proof theory symposium, Kiel 1974","volume":"500","author":"Buchholz","year":"1975"},{"key":"S0022481200011968_ref004","volume-title":"Annals of Pure and Applied Logic","author":"Arai"},{"key":"S0022481200011968_ref003","volume-title":"Archive for Mathematical Logic","author":"Arai"},{"key":"S0022481200011968_ref002","unstructured":"Arai T. , A system (O(\u03c03), <) of ordinal diagrams, handwritten note, October 1990."},{"key":"S0022481200011968_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF01625829"},{"key":"S0022481200011968_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/421108"},{"key":"S0022481200011968_ref001","unstructured":"Arai T. , Proof theory for reflecting ordinals III: \u03a03-reflecting ordinals, handwritten note, December 1990."},{"key":"S0022481200011968_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02023014"},{"key":"S0022481200011968_ref010","volume-title":"Annals of Pure and Applied Logic","author":"Schl\u00fcter"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200011968","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T20:47:18Z","timestamp":1557348438000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200011968\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2000,9]]}},"alternative-id":["S0022481200011968"],"URL":"https:\/\/doi.org\/10.2307\/2586705","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,9]]}}}