{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T20:16:29Z","timestamp":1775506589343,"version":"3.50.1"},"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>Given <jats:italic>I<\/jats:italic>, a reasonable countable set of Turing degrees, can we find some sort of canonical strict upper bound on <jats:italic>I<\/jats:italic>? If <jats:italic>I<\/jats:italic> = {<jats:underline>a<\/jats:underline> \u2223 <jats:underline>a<\/jats:underline> \u2264 <jats:underline>b<\/jats:underline>}, the upper bound on <jats:italic>I<\/jats:italic> which springs to mind is <jats:underline>b<\/jats:underline>\u2032. But what if <jats:italic>I<\/jats:italic> is closed under jump? This question arises naturally out of the question which motivates a large part of hierarchy theory: Is there a canonical increasing function from a countable ordinal, preferably a large one, into <jats:italic>D<\/jats:italic>, the set of Turing degrees? If <jats:italic>d<\/jats:italic> is to be such a function, it is natural to require that <jats:italic>d<\/jats:italic>(\u03b1 + 1) = <jats:italic>d<\/jats:italic>(\u03b1)\u2032; but how should <jats:italic>d<\/jats:italic>(\u03bb) depend on <jats:italic>d<\/jats:italic> \u21be \u03bb, where \u03bb is a limit ordinal?<\/jats:p><jats:p>For any <jats:italic>I<\/jats:italic> \u2286 <jats:italic>D<\/jats:italic>, let <jats:italic>M<\/jats:italic><jats:sub><jats:italic>I<\/jats:italic><\/jats:sub>, = \u22c3<jats:italic>I<\/jats:italic>. Towards making the above questions precise, we introduce ideals of Turing degrees.<\/jats:p><jats:p>Definition 1. <jats:italic>I \u2286 D<\/jats:italic> is an ideal iff <jats:italic>I<\/jats:italic> is closed under jump and join, and <jats:italic>I<\/jats:italic> is downward-closed, i.e., if <jats:underline><jats:italic>a<\/jats:italic><\/jats:underline> \u2264 <jats:underline><jats:italic>b<\/jats:italic><\/jats:underline> &amp; <jats:underline><jats:italic>b<\/jats:italic><\/jats:underline> \u03f5 <jats:italic>I<\/jats:italic> then <jats:italic><jats:underline>a<\/jats:underline><\/jats:italic> \u03f5 <jats:italic>I<\/jats:italic>.<\/jats:p><jats:p>The following definition reflects the hierarchy-theoretic motivation for this paper.<\/jats:p><jats:p>Definition 2. For <jats:italic>I<\/jats:italic> \u2286 <jats:italic>D<\/jats:italic> and <jats:italic>A<\/jats:italic> \u2286 \u03c9, <jats:italic>I<\/jats:italic> is an <jats:italic>A<\/jats:italic>-hierarchy ideal iff for some countable ordinal \u03b1, <jats:italic>M<\/jats:italic><jats:sub><jats:italic>I<\/jats:italic><\/jats:sub> = <jats:italic>L<jats:sub>\u03b1<\/jats:sub>[A]\u2229 \u03c9<jats:sup>\u03c9<\/jats:sup><\/jats:italic>.<\/jats:p><jats:p>All hierarchy ideals are ideals, but not conversely.<\/jats:p><jats:p>Early in the game Spector knocked out the best sort of canonicity for upper bounds on ideals, proving that no set of degrees closed under jump has a least upper bound.<\/jats:p>","DOI":"10.2307\/2273535","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:48:21Z","timestamp":1146937701000},"page":"601-612","source":"Crossref","is-referenced-by-count":2,"title":["Uniform upper bounds on ideals of turing degrees"],"prefix":"10.1017","volume":"43","author":[{"given":"Harold T.","family":"Hodes","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004946X_ref005","volume-title":"The theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S002248120004946X_ref004","volume-title":"Logic Colloquium '69","author":"Leeds"},{"key":"S002248120004946X_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90023-1"},{"key":"S002248120004946X_ref002","first-page":"64","volume":"30","author":"Hensel","year":"1965","journal-title":"On the notational independence of various hierarchies of degrees of unsolvability"},{"key":"S002248120004946X_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1969-0242673-X"},{"key":"S002248120004946X_ref006","volume-title":"Mathematical logic","author":"Shoenfield","year":"1967"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004946X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:33:38Z","timestamp":1558971218000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004946X\/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":["S002248120004946X"],"URL":"https:\/\/doi.org\/10.2307\/2273535","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}