{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T11:29:56Z","timestamp":1648639796920},"reference-count":3,"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>We present two theorems whose applications are to eliminate diagonalization arguments from a variety of constructions of degrees of unsolvability.<\/jats:p><jats:p>All definitions and notations come from [1, Chapter 1]. We give a brief resum\u00e9 of them here.<\/jats:p><jats:p>We identify a set with its characteristic function. (<jats:italic>A<\/jats:italic>(<jats:italic>x<\/jats:italic>)= 1 if <jats:italic>x<\/jats:italic> \u2208 <jats:italic>A<\/jats:italic> and <jats:italic>A<\/jats:italic> (<jats:italic>x<\/jats:italic>) = 0 if <jats:italic>x<\/jats:italic> \u2209 <jats:italic>A<\/jats:italic>.) A string \u03c3 is the restriction of a characteristic function to a finite initial segment of natural numbers, lh(\u03c3) = length of \u03c3 = <jats:italic>n<\/jats:italic> + 1 if \u03c3 = <jats:italic>A<\/jats:italic>[<jats:italic>n<\/jats:italic>] for some set <jats:italic>A<\/jats:italic>. (<jats:italic>A<\/jats:italic> [<jats:italic>n<\/jats:italic>] is the restriction of <jats:italic>A<\/jats:italic> to {<jats:italic>m<\/jats:italic>: <jats:italic>m<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic>}.) If <jats:italic>i<\/jats:italic> = 0 or 1, \u03c3 * <jats:italic>i<\/jats:italic> is defined as the string of length lh(\u03c3) + 1 such that \u03c3 * <jats:italic>i<\/jats:italic> \u2287 \u03c3 and \u03c3 * <jats:italic>i<\/jats:italic>(lh(\u03c3)) = <jats:italic>i<\/jats:italic>. We write \u03c3 \u2223 \u03c4 if \u03c3 \u2289 \u03c4 and \u03c4 \u2289 \u03c3.<\/jats:p><jats:p>{\u03a6<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>} is a listing of the partial recursive functionals. We write \u201c<jats:italic>A<\/jats:italic> \u2264<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub><jats:italic>B<\/jats:italic>\u201d (\u201c<jats:italic>A<\/jats:italic> is Turing reducible to <jats:italic>B<\/jats:italic>\u201d) if \u2203<jats:italic>n<\/jats:italic>\u2200<jats:italic>x<\/jats:italic>\u03a6<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>B<\/jats:italic>)(<jats:italic>x<\/jats:italic>) = <jats:italic>A<\/jats:italic>(<jats:italic>x<\/jats:italic>).<\/jats:p><jats:p>A partial function, <jats:italic>T<\/jats:italic>, from strings to strings is a tree if <jats:italic>T<\/jats:italic> is order preserving and for all strings, \u03c3, if one of <jats:italic>T<\/jats:italic>(\u03c3 * 0), <jats:italic>T<\/jats:italic>(\u03c3 * 1) is defined then <jats:italic>T<\/jats:italic>(\u03c3), <jats:italic>T<\/jats:italic>(\u03c3 * 0), <jats:italic>T<\/jats:italic>(\u03c3 * 1) are all defined and <jats:italic>T<\/jats:italic>(\u03c3 * 0)\u2223(\u03c3 * 1).<\/jats:p>","DOI":"10.2307\/2272826","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:47:35Z","timestamp":1146952055000},"page":"280-283","source":"Crossref","is-referenced-by-count":5,"title":["Diagonalization in degree constructions"],"prefix":"10.1017","volume":"43","author":[{"given":"D.","family":"Posner","sequence":"first","affiliation":[]},{"given":"R.","family":"Epstein","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049628_ref001","volume-title":"Memoirs of the American Mathematical Society","author":"Epstein","year":"1975"},{"key":"S0022481200049628_ref003","first-page":"243","volume":"35","author":"Yates","year":"1970","journal-title":"Initial segments of the degrees of unsolvability, Part II: Minimal degrees"},{"key":"S0022481200049628_ref002","first-page":"289","volume":"41","author":"Lachlan","year":"1976","journal-title":"Countable initial segments of the degrees of unsolvability"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049628","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:57:38Z","timestamp":1558987058000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049628\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049628"],"URL":"https:\/\/doi.org\/10.2307\/2272826","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}