{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T21:45:59Z","timestamp":1698270359777},"reference-count":4,"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":15167,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,9]]},"abstract":"<jats:p>An infinite set <jats:italic>A<\/jats:italic> is said to be <jats:italic>hyperhyperimmune<\/jats:italic> (h.h.i.) if, for any collection of disjoint simultaneously recursively enumerable (r.e.) finite sets, <jats:italic>A<\/jats:italic> must fail to intersect with one of those sets. Thus the elements of an h.h.i. set are, in a sense, very elusive. D. A. Martin [3] showed that the degrees of h.h.i. sets with r.e. complements are exactly the r.e. degrees with jump <jats:bold>0\u2033<\/jats:bold>. More generally, C. G. Jockusch [2] found <jats:bold>a\u2032<\/jats:bold> \u2265 <jats:bold>0\u2033<\/jats:bold> to be a sufficient condition for <jats:bold>a<\/jats:bold> to be the degree of an h.h.i. set and found <jats:bold>a\u2032<\/jats:bold> \u2265 <jats:bold>0\u2032<\/jats:bold> to be necessary. However, it was also shown that in the degrees as a whole neither condition gave a characterization of the h.h.i. degrees. The purpose of this note is to prove that <jats:bold>a\u2032<\/jats:bold> = <jats:bold>0\u2033<\/jats:bold><jats:italic>does<\/jats:italic> characterize the h.h.i. degrees below <jats:bold>0\u2032<\/jats:bold>.<\/jats:p><jats:p>Theorem. <jats:italic>The degrees below <jats:bold>0\u2032<\/jats:bold> containing h.h.i. sets are exactly those degrees below <jats:bold>0\u2032<\/jats:bold> with jump <jats:bold>0\u2033<\/jats:bold><\/jats:italic>.<\/jats:p><jats:p>Proof. From [2], if <jats:bold>a\u2032<\/jats:bold> \u2265 <jats:bold>0\u2033<\/jats:bold>, then a contains an h.h.i. set.<\/jats:p><jats:p>Conversely, let <jats:italic>A<\/jats:italic> \u2208 <jats:bold>a<\/jats:bold> where <jats:bold>a\u2032<\/jats:bold> &lt; <jats:bold>0\u2033<\/jats:bold> and <jats:bold>a<\/jats:bold> &lt; <jats:bold>0\u2032<\/jats:bold>. Let {<jats:italic>A<jats:sub>s<\/jats:sub><\/jats:italic> \u2223 <jats:italic>s<\/jats:italic> \u2265 0} be a recursive sequence of finite sets such that for each <jats:italic>x<\/jats:italic>, lim<jats:sub><jats:italic>s<\/jats:italic><\/jats:sub>, <jats:italic>A<jats:sub>x<\/jats:sub><\/jats:italic>(<jats:italic>x<\/jats:italic>) exists and equals <jats:italic>A<\/jats:italic>(<jats:italic>x<\/jats:italic>).<\/jats:p><jats:p>For a set <jats:italic>B<\/jats:italic>, let <jats:italic>B<\/jats:italic>[<jats:italic>m<\/jats:italic>] denote <jats:italic>B<\/jats:italic> \u2229 [0, <jats:italic>m<\/jats:italic>], and (if <jats:italic>B<\/jats:italic> is finite) let \u2223<jats:italic>B<\/jats:italic>\u2223 denote the cardinality of <jats:italic>B<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2272750","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:17:55Z","timestamp":1146935875000},"page":"598-600","source":"Crossref","is-referenced-by-count":11,"title":["Jump equivalence of the \u0394<sub>2<\/sub><sup>0<\/sup> hyperimmune sets"],"prefix":"10.1017","volume":"37","author":[{"given":"S. B.","family":"Cooper","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200079196_ref001","unstructured":"Cooper S. B. , Minimal degrees and the jump-operator, this Journal (to appear)."},{"key":"S0022481200079196_ref003","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"S0022481200079196_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19620080313"},{"key":"S0022481200079196_ref002","first-page":"489","volume":"34","author":"Jockusch","year":"1969","journal-title":"The degrees of hyperhyperimmune sets"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200079196","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T16:58:15Z","timestamp":1559235495000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200079196\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1972,9]]}},"alternative-id":["S0022481200079196"],"URL":"https:\/\/doi.org\/10.2307\/2272750","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,9]]}}}