{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T15:14:59Z","timestamp":1672845299217},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11607,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,6]]},"abstract":"<jats:p>It is known that every &lt; <jats:sub>\u03b50<\/jats:sub>-recursive function is also a primitive recursive functional. Kreisel has proved this by means of G\u00f6del's functional-interpretation, using that every &lt; <jats:sub>\u03b50<\/jats:sub>-recursive function is provably recursive in Heyting's arithmetic [2, \u00a73.4]. Parsons obtained a refinement of Kreisel's result by a further examination of G\u00f6del's interpretation with regard to type levels [3, Theorem 5], [4, \u00a74]. A quite different proof is provided by the research into extensions of the Grzegorczyk hierarchy as done by Schwichtenberg and Wainer: this yields another characterization of the &lt; <jats:sub>\u03b50<\/jats:sub>-recursive functions from which easily appears that these are primitive recursive functionals (see [5] in combination with [6, Chapter II]).<\/jats:p><jats:p>However, these proofs are indirect and do not show how, in general, given a definition tree of an ordinal recursive functional, transfinite recursions can be replaced (in a straightforward way) by recursions over wellorderings of lower order types. The argument given by Tait in [9, pp. 189\u2013191] seems to be an improvement in this respect, but the crucial step in it is (at least in my opinion) not very clear.<\/jats:p>","DOI":"10.2307\/2273149","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:59:42Z","timestamp":1146938382000},"page":"395-402","source":"Crossref","is-referenced-by-count":9,"title":["On definition trees of ordinal recursive functionals: Reduction of the recursion orders by means of type level raising"],"prefix":"10.1017","volume":"47","author":[{"given":"Jan","family":"Terlouw","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044315_ref010","first-page":"281","volume":"37","author":"Wainer","year":"1972","journal-title":"Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy"},{"key":"S0022481200044315_ref007","first-page":"279","volume-title":"Logic Colloquium '73","author":"Schwichtenberg","year":"1975"},{"key":"S0022481200044315_ref003","first-page":"361","volume":"36","author":"Parsons","year":"1971","journal-title":"Proof-theoretic analysis of restricted induction schemata (abstract)"},{"key":"S0022481200044315_ref002","first-page":"101","volume-title":"Constructivity in mathematics","author":"Kreisel","year":"1959"},{"key":"S0022481200044315_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01564760"},{"key":"S0022481200044315_ref009","first-page":"185","volume-title":"Logic, methodology and the philosophy of science. III","author":"Tait","year":"1967"},{"key":"S0022481200044315_ref005","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19710170113"},{"key":"S0022481200044315_ref004","first-page":"466","volume":"37","author":"Parsons","year":"1972","journal-title":"On n-quantifier induction"},{"key":"S0022481200044315_ref006","volume-title":"Einige Anwendungen von Unendlichen Ternten und Wertfunktionalen","author":"Schwichtenberg","year":"1973"},{"key":"S0022481200044315_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF01342980"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044315","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T17:16:22Z","timestamp":1558718182000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044315\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,6]]},"references-count":10,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1982,6]]}},"alternative-id":["S0022481200044315"],"URL":"https:\/\/doi.org\/10.2307\/2273149","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,6]]}}}