{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T18:08:01Z","timestamp":1726250881531},"reference-count":4,"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":14072,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,9]]},"abstract":"<jats:p>Whenever particular ordinals are used as tools in a proof or a definition, it is necessary to find a way of representing them. If the ordinals are sufficiently small, there is a standard way (e.g. Cantor normal forms for ordinals less than <jats:italic>\u03b5<\/jats:italic><jats:sub>0<\/jats:sub>); in general, representations are often found by using functions on initial segments of the ordinals: Each term which can be obtained from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052932_inline1\" \/> by applications of a function symbol <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052932_inline2\" \/> is regarded as a notation for the ordinal obtained by the same applications of the function <jats:italic>f<\/jats:italic> to the ordinal 0. In this way, <jats:italic>f<\/jats:italic> provides representations for all the ordinals in Cl<jats:sub><jats:italic>f<\/jats:italic><\/jats:sub>(0), the closure set of <jats:italic>f<\/jats:italic> (se e \u00a71). (For an introduction to and development of this principle, see Feferman [F1]; and for a discussion of the significance of such representations in proof theory, see Kreisel [K1, pp. 22\u201334].) Thus it is natural to ask whether there are connections between frequently encountered properties of ordinal functions and the size of the ordinals for which they can provide representations.<\/jats:p><jats:p>The purpose of this paper is to show that, for any integer <jats:italic>n<\/jats:italic>, the ordinal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052932_inline3\" \/> (see \u00a72) is a bound for the closure ordinals of replete monotonic increasing <jats:italic>n<\/jats:italic>-place functions. This result is optimal for <jats:italic>n<\/jats:italic> &gt; 2 (the bound is attained by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052932_inline4\" \/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052932_inline5\" \/> where <jats:italic>\u03b8\u03b1<\/jats:italic> = 1 + <jats:italic>\u03b1<\/jats:italic>) but not for <jats:italic>n<\/jats:italic> &lt; 2. D. H. de Jongh has recently proved, using a completely different method, that the (least possible) bound for <jats:italic>n<\/jats:italic> = 2 is \u03b5<jats:sub>0<\/jats:sub>. (Trivially, that for <jats:italic>n<\/jats:italic> = 1 is <jats:italic>\u03c6<\/jats:italic>.)<\/jats:p>","DOI":"10.2307\/2272156","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:36:35Z","timestamp":1146951395000},"page":"305-316","source":"Crossref","is-referenced-by-count":15,"title":["Bounds for the closure ordinals of replete monotonic increasing functions"],"prefix":"10.1017","volume":"40","author":[{"given":"Diana","family":"Schmidt","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200052932_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01361109"},{"key":"S0022481200052932_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-88514-3"},{"key":"S0022481200052932_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70845-0"},{"key":"S0022481200052932_ref002","first-page":"193","volume":"33","author":"Feferman","year":"1968","journal-title":"Systems of predicative analysis. II: Representations of ordinals"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200052932","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T19:21:23Z","timestamp":1559157683000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200052932\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1975,9]]}},"alternative-id":["S0022481200052932"],"URL":"https:\/\/doi.org\/10.2307\/2272156","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,9]]}}}