{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T03:28:54Z","timestamp":1648524534772},"reference-count":6,"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":13068,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,6]]},"abstract":"<jats:p>Several new features arise in the generalization of recursion theory on \u03c9 to recursion theory on admissible ordinals \u03b1, thus making \u03b1-recursion theory an interesting theory. One of these is the appearance of irregular sets. A subset <jats:italic>A<\/jats:italic> of \u03b1 is called regular (over \u03b1), if we have for all \u03b2 &lt; \u03b1 that <jats:italic>A<\/jats:italic> \u2229 <jats:italic>B<\/jats:italic> \u2208 <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub>, otherwise <jats:italic>A<\/jats:italic> is called irregular (over \u03b1). So in the special case of ordinary recursion theory (\u03b1 = \u03c9) every subset of \u03b1 is regular, but if \u03b1 is not a cardinal of <jats:italic>L<\/jats:italic> we find constructible sets <jats:italic>A<\/jats:italic> \u2286 \u03b1 which are irregular. The notion of regularity becomes essential, if we deal with \u03b1-recursively enumerable (\u03b1-r.e.) sets in priority constructions (\u03b1-r.e. is defined as \u03a3<jats:sub>1<\/jats:sub> over <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub>). The typical situation occurring there is that an \u03b1-r.e. set <jats:italic>A<\/jats:italic> is enumerated during some construction in which one tries to satisfy certain requirements. Often this construction succeeds only if we can insure that every initial segment <jats:italic>A<\/jats:italic> \u2229 \u03b2 of <jats:italic>A<\/jats:italic> is completely enumerated at some stage before \u03b1. This calls for making sure that <jats:italic>A<\/jats:italic> is regular because due to the admissibility of \u03b1 an \u03b1-r.e. set <jats:italic>A<\/jats:italic> is regular iff for every (or equivalently for one) enumeration <jats:italic>f<\/jats:italic> of <jats:italic>A<\/jats:italic> (<jats:italic>f<\/jats:italic> is an enumeration of <jats:italic>A<\/jats:italic> iff <jats:italic>f<\/jats:italic>: \u03b1 \u2192 <jats:italic>A<\/jats:italic> is \u03b1-recursive, total, 1-1 and onto) we have that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049616_inline1\" \/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049616_inline2\" \/> is the image of the set \u03c3 under <jats:italic>f<\/jats:italic>).<\/jats:p>","DOI":"10.2307\/2272825","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:47:35Z","timestamp":1146952055000},"page":"270-279","source":"Crossref","is-referenced-by-count":2,"title":["The uniform regular set theorem in \u03b1-recursion theory"],"prefix":"10.1017","volume":"43","author":[{"given":"Wolfgang","family":"Maass","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049616_ref006","first-page":"295","volume":"39","author":"Shore","year":"1974","journal-title":"\u03a3n sets which are \u0394n-incomparable (uniformly)"},{"key":"S0022481200049616_ref005","first-page":"65","article-title":"Splitting an \u03b1-recursively enumerable set","volume":"204","author":"Shore","year":"1975","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049616_ref004","unstructured":"Simpson S. G. , Admissible ordinals and recursion theory, Thesis, MIT, 1974."},{"key":"S0022481200049616_ref002","volume-title":"Annals of Mathematical Logic","author":"Maass"},{"key":"S0022481200049616_ref001","volume-title":"Springer Lecture Notes","author":"Devlin","year":"1973"},{"key":"S0022481200049616_ref003","first-page":"1","article-title":"Post's problem, admissible ordinals and regularity","volume":"124","author":"Sacks","year":"1966","journal-title":"Transactions of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049616","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:57:39Z","timestamp":1558987059000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049616\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049616"],"URL":"https:\/\/doi.org\/10.2307\/2272825","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}