{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,14]],"date-time":"2022-06-14T11:13:22Z","timestamp":1655205202429},"reference-count":6,"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":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>In [5] S. S. Wainer introduces a hierarchy for arbitrary type-2-functionals. Given <jats:italic>F<\/jats:italic>, he defines a set of ordinal notations <jats:italic>O<jats:sup>F<\/jats:sup><\/jats:italic>, and for each <jats:italic>a<\/jats:italic> \u2208 <jats:italic>O<jats:sup>F<\/jats:sup><\/jats:italic>, a function <jats:italic>f<jats:sub>a<\/jats:sub><\/jats:italic> recursive in <jats:italic>F<\/jats:italic> and an ordinal \u2223<jats:italic>a<\/jats:italic>\u2223<jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> &lt; <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049367_inline1\" \/> For any <jats:italic>f<\/jats:italic> recursive in <jats:italic>F<\/jats:italic> there is an <jats:italic>a<\/jats:italic> \u2208 <jats:italic>O<jats:sup>F<\/jats:sup><\/jats:italic> such that <jats:italic>f<\/jats:italic> is primitive recursive in <jats:italic>f<jats:sub>a<\/jats:sub><\/jats:italic>.<\/jats:p><jats:p>Let \u03c1<jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> be the least ordinal \u03b1 such that for any <jats:italic>f<\/jats:italic> recursive in <jats:italic>F<\/jats:italic> there is an \u03b1 \u2208 <jats:italic>O<jats:sup>F<\/jats:sup><\/jats:italic> with \u2223<jats:italic>a<\/jats:italic>\u2223<jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> \u2264 \u03b1 such that <jats:italic>f<\/jats:italic> is primitive recursive in <jats:italic>f<jats:sub>a<\/jats:sub><\/jats:italic>. If \u03c1<jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> &lt; <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049367_inline2\" \/> the hierarchy breaks down. In Bergstra and Wainer [2] \u03c1<jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> is described as \u201cthe real ordinal of the 1-section of <jats:italic>F<\/jats:italic>\u201d.<\/jats:p><jats:p>Using standard methods (originally due to Kleene) one may prove that if <jats:italic>F<\/jats:italic> is normal, then \u03c1<jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> = <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049367_inline1\" \/> Feferman has proved that if <jats:italic>F<\/jats:italic> is recursive, then \u03c1<jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> = \u03c9<jats:sup>2<\/jats:sup>.<\/jats:p><jats:p>Let 1-section (<jats:italic>F<\/jats:italic>) = l-sc(<jats:italic>F<\/jats:italic>) = {<jats:italic>f<\/jats:italic>; <jats:italic>f<\/jats:italic> is recursive in <jats:italic>F<\/jats:italic>} where <jats:italic>f<\/jats:italic> is a total object of type 1.<\/jats:p><jats:p>Grilliot [4] proved that <jats:italic>F<\/jats:italic> \u21be 1-sc(<jats:italic>F<\/jats:italic>) is continuous if and only if <jats:italic>F<\/jats:italic> is not normal.<\/jats:p><jats:p>Let <jats:italic>h<\/jats:italic> be an associate for a given functional <jats:italic>F<\/jats:italic>, and assume that <jats:italic>h<\/jats:italic> is recursive in the jump of an element of 1-sc(<jats:italic>F<\/jats:italic>).<\/jats:p>","DOI":"10.2307\/2273524","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:48:21Z","timestamp":1146937701000},"page":"487-491","source":"Crossref","is-referenced-by-count":1,"title":["A continuous functional with noncollapsing hierarchy"],"prefix":"10.1017","volume":"43","author":[{"given":"Dag","family":"Normann","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049367_ref005","first-page":"88","volume":"39","author":"Wainer","year":"1974","journal-title":"A hierarchy for the 1-section of any type two object"},{"key":"S0022481200049367_ref001","unstructured":"Bergstra J. , Computability and continuity in finite types, Dissertation, Utrecht, 1976."},{"key":"S0022481200049367_ref002","first-page":"440","volume":"42","author":"Bergstra","year":"1977","journal-title":"The \u201creal\u201d ordinal of the 1-section of a continuous functional"},{"key":"S0022481200049367_ref003","first-page":"571","article-title":"Proof of Mostowski's conjecture","volume":"9","author":"Gandy","year":"1960","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences"},{"key":"S0022481200049367_ref006","volume-title":"Annals of mathematics studies","author":"Sacks","year":"1963"},{"key":"S0022481200049367_ref004","first-page":"245","volume":"36","author":"Grilliot","year":"1971","journal-title":"On effectively discontinuous type-2 objects"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049367","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:33:51Z","timestamp":1558971231000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049367\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S0022481200049367"],"URL":"https:\/\/doi.org\/10.2307\/2273524","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}