{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T16:56:39Z","timestamp":1648745799413},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":15351,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,3]]},"abstract":"<jats:p>Bachmann, in [2] shows how certain ordinals &lt;<jats:italic>\u03a9<\/jats:italic>(<jats:italic>\u03a9<\/jats:italic> = <jats:italic>\u03a9<\/jats:italic><jats:sub>1<\/jats:sub> where <jats:italic>\u03a9<\/jats:italic><jats:sub><jats:italic>\u03be<\/jats:italic><\/jats:sub> is the (1 + <jats:italic>\u03be<\/jats:italic>)th infinite initial ordinal) may be described from below using suitable descriptions of ordinals &lt;<jats:italic>\u03a9<\/jats:italic><jats:sub>2<\/jats:sub>. The aim of this paper is to consider another approach to describing ordinal &lt;<jats:italic>\u03a9<\/jats:italic> and compare it with the Bachmann method. Our approach will use functionals of transfinite type based on <jats:italic>\u03a9<\/jats:italic>.<\/jats:p><jats:p>The Bachmann method consists in denning a hierarchy <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline1\" \/> of normal functions <jats:italic>\u03d5<\/jats:italic><jats:sub><jats:italic>\u03b4<\/jats:italic><\/jats:sub>: <jats:italic>\u03a9<\/jats:italic> \u2192 <jats:italic>\u03a9<\/jats:italic> (i.e. continuous and strictly increasing) for <jats:italic>\u03b4<\/jats:italic> \u2264 <jats:italic>\u03b7<\/jats:italic><jats:sub>0<\/jats:sub> &lt; <jats:italic>\u03a9<\/jats:italic><jats:sub>2<\/jats:sub>, starting with <jats:italic>\u03d5<\/jats:italic><jats:sub>0<\/jats:sub>(<jats:italic>\u03bb<\/jats:italic>) = <jats:italic>\u03c9<\/jats:italic><jats:sup>1 + <jats:italic>\u03bb<\/jats:italic><\/jats:sup>. The definition of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline2\" \/> depends on a suitable description of the ordinals \u2264 \u03b7<jats:sub>0<\/jats:sub>. This is obtained by defining a hierarchy \u3008<jats:italic>F<\/jats:italic><jats:sub><jats:italic>\u03b4<\/jats:italic><\/jats:sub> \u2223 \u03b4 \u2264 <jats:italic>\u03a9<\/jats:italic><jats:sub>2<\/jats:sub>\u3009 of normal functions <jats:italic>F<\/jats:italic><jats:sub><jats:italic>\u03b4<\/jats:italic><\/jats:sub>: <jats:italic>\u03a9<\/jats:italic><jats:sub>2<\/jats:sub> \u2192 <jats:italic>\u03a9<\/jats:italic><jats:sub>2<\/jats:sub> analogously to the definition of the initial segment \u3008\u03d5<jats:sub><jats:italic>\u03b4<\/jats:italic><\/jats:sub> \u2223 <jats:italic>\u03b4<\/jats:italic> \u2264 <jats:italic>\u03a9<\/jats:italic>\u3009 of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline2\" \/>. The ordinal <jats:italic>\u03b7<\/jats:italic><jats:sub>0<\/jats:sub> is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline3\" \/>.<\/jats:p><jats:p><jats:italic>Note<\/jats:italic>. Our description of Bachmann's hierarchies will differ slightly from those in Bachmann's paper. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline4\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline5\" \/> denote the hierarchies in [2]. Then as Bachmann's normal functions are not defined at 0 we let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline6\" \/> for <jats:italic>\u03bb, \u03b4<\/jats:italic> &lt; <jats:italic>\u03a9<\/jats:italic><jats:sub>2<\/jats:sub>. Bachmann defines <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline7\" \/> for 0 &lt; <jats:italic>\u03bb<\/jats:italic> &lt; <jats:italic>\u03a9<\/jats:italic><jats:sub>2<\/jats:sub> but it seems more natural to omit this so that we let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline8\" \/>. The situation is analogous for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline9\" \/> and leads to the following definitions:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_inline10\" \/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic> and <jats:italic>\u03be<\/jats:italic> is a limit number of cofinality <jats:italic>\u03a9<\/jats:italic>, and<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120008052X_eqnU2\" \/><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2272543","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:14:18Z","timestamp":1146950058000},"page":"35-47","source":"Crossref","is-referenced-by-count":9,"title":["Describing ordinals using functionals of transfinite type"],"prefix":"10.1017","volume":"37","author":[{"given":"Peter","family":"Aczel","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120008052X_ref011","unstructured":"Pfeiffer H. , Ausgezetchnete Folgen fur gewisse Abschnitte der zweiten und weiterer Zahlklassen, Doctoral dissertation, Technische Hochschule, Hanover, 1964."},{"key":"S002248120008052X_ref010","first-page":"319","article-title":"Zur Konstruktion von Ordnungszahlen","volume":"58","author":"Neumer","year":"1953","journal-title":"Mathematische Zeitschrift. I"},{"key":"S002248120008052X_ref009","first-page":"339","volume-title":"Intuitionism and proof theory","author":"Isles","year":"1970"},{"key":"S002248120008052X_ref008","first-page":"339","volume-title":"Intuitionism and proof theory","author":"Gerber","year":"1970"},{"key":"S002248120008052X_ref007","first-page":"303","volume-title":"Intuitionism and proof theory","author":"Feferman","year":"1970"},{"key":"S002248120008052X_ref006","first-page":"289","volume-title":"Intuitionism and proof theory","author":"Feferman","year":"1970"},{"key":"S002248120008052X_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71190-X"},{"key":"S002248120008052X_ref002","first-page":"5","article-title":"Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen","volume":"95","author":"Bachmann","year":"1950","journal-title":"Vierteljahrschrift der Naturforscherden Gesellschaft in Z\u00fcrich"},{"key":"S002248120008052X_ref012","volume-title":"Predicative well-orderings, Formal systems and recursive functions","author":"Sch\u00fctte","year":"1965"},{"key":"S002248120008052X_ref001","unstructured":"Aczel P. , Three systems of notations for ordinals (unpublished), 1969."},{"key":"S002248120008052X_ref004","first-page":"193","volume":"33","author":"Feferman","year":"1964","journal-title":"Systems of predicative analysis, II: Representations of ordinals"},{"key":"S002248120008052X_ref003","first-page":"1","volume":"29","author":"Feferman","year":"1964","journal-title":"Systems of predicative analysis"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120008052X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T19:18:08Z","timestamp":1559330288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120008052X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,3]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1972,3]]}},"alternative-id":["S002248120008052X"],"URL":"https:\/\/doi.org\/10.2307\/2272543","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,3]]}}}