{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T14:06:08Z","timestamp":1775829968354,"version":"3.50.1"},"reference-count":13,"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":14802,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1973,9]]},"abstract":"<jats:p>Let <jats:italic>\u03c9<\/jats:italic> be the nonnegative integers. G. E. Sacks once asked whether there exists an infinite <jats:italic>X<\/jats:italic> \u2286 <jats:italic>\u03c9<\/jats:italic> such that, for all <jats:italic>Y<\/jats:italic> \u2286 <jats:italic>X<\/jats:italic>, <jats:italic>\u03c9<\/jats:italic><jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>Y<\/jats:italic><\/jats:sup><jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> where <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> is the first nonrecursive ordinal. In this note we negatively answer this question by giving a simple proof that for every infinite set <jats:italic>X<\/jats:italic> \u2286 <jats:italic>\u03c9<\/jats:italic> there exists <jats:italic>Y<\/jats:italic> \u2286 <jats:italic>X<\/jats:italic> such that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200054803_inline1\"\/> the first recursively inaccessible ordinal. This is accomplished by proving that <jats:italic>H<jats:sub>\u03b1<\/jats:sub><\/jats:italic> is hyper-arithmetic in <jats:italic>Y<\/jats:italic> where<jats:italic>H<jats:sub>\u03b1<\/jats:sub><\/jats:italic> is the <jats:italic>\u03b1<\/jats:italic>th hyperjump of the empty set \u2205, defined in a suitable sense for all ordinals <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200054803_inline2\"\/><\/jats:p><jats:p>Background information and undefined notation can be found in Rogers [11]. In particular, we write <jats:italic>A<\/jats:italic> \u2264<jats:sub><jats:italic>h<\/jats:italic><\/jats:sub><jats:italic>B(A \u2264<jats:sub>T<\/jats:sub> B)<\/jats:italic> if <jats:italic>A<\/jats:italic> is hyperarithmetical (recursive) in <jats:italic>B<\/jats:italic>, and <jats:italic>A<\/jats:italic> \u2261<jats:sub><jats:italic>h<\/jats:italic><\/jats:sub><jats:italic>B<\/jats:italic> if <jats:italic>A<\/jats:italic> \u2264<jats:sub><jats:italic>h<\/jats:italic><\/jats:sub><jats:italic>B<\/jats:italic> and <jats:italic>B<\/jats:italic> \u2264<jats:sub><jats:italic>h<\/jats:italic><\/jats:sub><jats:italic>A<\/jats:italic>. We will say that a set <jats:italic>A<\/jats:italic> is hyperarithmetically (recursively) encodable if, for every infinite set <jats:italic>X<\/jats:italic> \u2286 <jats:italic>\u03c9<\/jats:italic>, there exists <jats:italic>Y<\/jats:italic> \u2286 <jats:italic>X<\/jats:italic> such that <jats:italic>A<\/jats:italic> \u2264<jats:sub><jats:italic>h<\/jats:italic><\/jats:sub><jats:italic>Y (A<\/jats:italic> \u2264<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub><jats:italic>Y<\/jats:italic>). For any set <jats:italic>A<\/jats:italic> (hyperdegree <jats:bold>a<\/jats:bold>) let <jats:italic>A<\/jats:italic>\u2032 (<jats:bold>a<\/jats:bold>\u2032) denote the hyperjump of <jats:italic>A<\/jats:italic> (<jats:bold>a<\/jats:bold>). Let <jats:bold>0<\/jats:bold> denote the hyperdegree of \u2205. A function <jats:italic>f majorizes<\/jats:italic> a function <jats:italic>g<\/jats:italic> if <jats:italic>f(n)<\/jats:italic> \u2265 <jats:italic>g(n)<\/jats:italic> for every <jats:italic>n<\/jats:italic>. <jats:bold>E<\/jats:bold><jats:sub>1<\/jats:sub> is the representing (type-2) functional of<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200054803_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>introduced by Tugu\u00e9 [13] (also Kleene [6]). Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200054803_inline3\"\/> be the smallest ordinal which is not the order type of any well-ordering recursive in <jats:bold>E<\/jats:bold><jats:sub>1<\/jats:sub>. Information on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200054803_inline3\"\/> can be found in Richter [9] and [10].<\/jats:p>","DOI":"10.2307\/2273040","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:24:44Z","timestamp":1146936284000},"page":"437-440","source":"Crossref","is-referenced-by-count":5,"title":["Encodability of Kleene's <i>O<\/i>"],"prefix":"10.1017","volume":"38","author":[{"suffix":"Jr.","given":"Carl G.","family":"Jockusch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert I.","family":"Soare","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054803_ref013","first-page":"97","article-title":"Predicates recursive in a type-2 object and Kleene hierarchies","volume":"8","author":"Tugu\u00e9","year":"1960","journal-title":"Commentarii Mathematici Universitatis Sancti Pauli"},{"key":"S0022481200054803_ref006","first-page":"106","article-title":"Recursive functionals and quantifiers of finite types, II","volume":"108","author":"Kleene","year":"1963","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200054803_ref003","unstructured":"Jockusch C. G. Jr. , Encodable sets (in preparation)."},{"key":"S0022481200054803_ref010","first-page":"43","volume":"33","author":"Richter","year":"1968","journal-title":"Constructively accessible ordinal numbers"},{"key":"S0022481200054803_ref001","first-page":"405","volume":"38","author":"Friedman","year":"1973","journal-title":"Borel sets and hyperdegrees"},{"key":"S0022481200054803_ref012","first-page":"53","volume":"34","author":"Soare","year":"1969","journal-title":"Sets with no subset of higher degree"},{"key":"S0022481200054803_ref002","first-page":"521","volume":"33","author":"Jockusch","year":"1968","journal-title":"Uniformly introreducible sets"},{"key":"S0022481200054803_ref007","first-page":"931","article-title":"On a generalization of Ramsey's theorem","volume":"15","author":"Mathias","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200054803_ref008","volume-title":"Lecture Notes in Mathematics","author":"Mathias"},{"key":"S0022481200054803_ref005","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1972.40.605"},{"key":"S0022481200054803_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1967-11710-5"},{"key":"S0022481200054803_ref011","volume-title":"Recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200054803_ref004","first-page":"33","article-title":"\u03a010 classes and degrees of theories","volume":"173","author":"Jockusch","year":"1972","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\/S0022481200054803","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T15:37:44Z","timestamp":1559230664000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054803\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1973,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1973,9]]}},"alternative-id":["S0022481200054803"],"URL":"https:\/\/doi.org\/10.2307\/2273040","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1973,9]]}}}