{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T06:10:52Z","timestamp":1683094252265},"reference-count":9,"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":10146,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>p<\/jats:italic> be a set. A function <jats:italic>\u03a6<\/jats:italic> is uniformly <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>p<\/jats:italic>) in every admissible set if there is a <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub> formula \u03d5 in the parameter <jats:italic>p<\/jats:italic> so that \u03d5 defines <jats:italic>\u03a6<\/jats:italic> in every <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub>-admissible set which includes <jats:italic>p<\/jats:italic>. A theorem of Van de Wiele states that if <jats:italic>\u03a6<\/jats:italic> is a total function from sets to sets then <jats:italic>\u03a6<\/jats:italic> is uniformly <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub> in every admissible set if and only if it is <jats:italic>E<\/jats:italic>-recursive. A function is <jats:italic>ES<jats:sub>p<\/jats:sub><\/jats:italic>-recursive if it can be generated from the schemes for <jats:italic>E<\/jats:italic>-recursion together with a selection scheme over the transitive closure of <jats:italic>p<\/jats:italic>. The selection scheme is exactly what is needed to insure that the <jats:italic>ES<jats:sub>P<\/jats:sub><\/jats:italic>-recursively enumerable predicates are closed under existential quantification over the transitive closure of <jats:italic>p<\/jats:italic>. Two theorems are established: a) If the transitive closure of <jats:italic>p<\/jats:italic> is countable then a total function on sets is <jats:italic>ES<jats:sub>p<\/jats:sub><\/jats:italic>-recursive if and only if it is uniformly <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>p<\/jats:italic>) in every admissible set. b) For any <jats:italic>p<\/jats:italic>, if <jats:italic>\u03a6<\/jats:italic> is a function on the ordinal numbers then <jats:italic>\u03a6<\/jats:italic> is <jats:italic>ES<jats:sub>P<\/jats:sub><\/jats:italic>-recursive if and only if it is uniformly <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>p<\/jats:italic>) in every admissible set.<\/jats:p>","DOI":"10.1017\/s0022481200031327","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T12:44:02Z","timestamp":1394714642000},"page":"453-461","source":"Crossref","is-referenced-by-count":0,"title":["<i>\u03a3<\/i><sub>1<\/sub> definitions with parameters"],"prefix":"10.1017","volume":"51","author":[{"given":"T. A.","family":"Slaman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200031327_ref009","volume-title":"Dilateurs r\u00e9cursifs et r\u00e9cursivit\u00e9s g\u00e9n\u00e9ralis\u00e9es","author":"van de Wiele"},{"key":"S0022481200031327_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/042\/791057"},{"key":"S0022481200031327_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/2373990"},{"key":"S0022481200031327_ref005","first-page":"303","volume-title":"Generalized recursion theory. 2","author":"Normann"},{"key":"S0022481200031327_ref002","unstructured":"Harrington L. A. , Contributions to recursion theory on higher types, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts."},{"key":"S0022481200031327_ref001","first-page":"5","article-title":"Generalized recursive functionals of finite type and hierarchies of functions","volume":"35","author":"Gandy","journal-title":"Annales de la Facult\u00e9 des Sciences de l'Universit\u00e9 de Clermont-Ferrand"},{"key":"S0022481200031327_ref003","unstructured":"Hoole M. R. R. , Recursion on sets, Ph.D. thesis, Wolfson College, Oxford."},{"key":"S0022481200031327_ref008","first-page":"1","article-title":"Reflection and forcing in E-recursion theory","volume":"28","author":"Slaman","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481200031327_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0419235-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031327","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T05:54:23Z","timestamp":1683093263000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031327\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,6]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1986,6]]}},"alternative-id":["S0022481200031327"],"URL":"https:\/\/doi.org\/10.1017\/s0022481200031327","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,6]]}}}