{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T06:10:57Z","timestamp":1683094257388},"reference-count":11,"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":10146,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic> be subsets of the reals. Say that <jats:italic>A<\/jats:italic><jats:sub><jats:italic>K<\/jats:italic><\/jats:sub>\u2265 <jats:italic>B<\/jats:italic>, if there is a real <jats:italic>a<\/jats:italic> such that the relation \u201c<jats:italic>x<\/jats:italic> \u2208 <jats:italic>B<\/jats:italic>\u201d is uniformly \u22bf<jats:sub>1<\/jats:sub> (<jats:italic>a, A<\/jats:italic>) in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline1\" \/>. This reducibility induces an equivalence relation \u2261<jats:sub><jats:italic>K<\/jats:italic><\/jats:sub> on the sets of reals; the \u2261<jats:sub><jats:italic>K<\/jats:italic><\/jats:sub>-equivalence class of a set is called its Kleene degree. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline2\" \/> be the structure that consists of the Kleene degrees and the induced partial order <jats:sub><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline2\" \/><\/jats:sub>\u2265. A substructure of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline2\" \/> that is of interest is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/>, the Kleene degrees of the <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline4\" \/> sets of reals. If sharps exist, then there is not much to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/>, as Steel [9] has shown that the existence of sharps implies that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/> has only two elements: the degree of the empty set and the degree of the complete <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline4\" \/> set. Legrand [4] used the hypothesis that there is a real whose sharp does not exist to show that there are incomparable elements in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/>; in the context of <jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic>, Hrb\u00e1\u010dek has shown that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/> is dense and has no minimal pairs. The Hrb\u00e1\u010dek results led Simpson [6] to make the following conjecture: if <jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic>, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/> forms a universal homogeneous upper semilattice with 0 and 1. Simpson's conjecture is shown to be false by showing that if <jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic>, then G\u00f6del's maximal thin <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline4\" \/> set is the infimum of two strictly larger elements of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/>.<\/jats:p><jats:p>The second main result deals with the notion of jump in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline2\" \/>. Let <jats:italic>A<\/jats:italic>\u2032 be the complete Kleene enumerable set relative to <jats:italic>A<\/jats:italic>. Say that <jats:italic>A<\/jats:italic> is low-<jats:italic>n<\/jats:italic> if <jats:italic>A<\/jats:italic><jats:sup>(<jats:italic>n<\/jats:italic>)<\/jats:sup> has the same degree as \u2298<jats:sup>(<jats:italic>n<\/jats:italic>)<\/jats:sup>, and <jats:italic>A<\/jats:italic> is high-<jats:italic>n<\/jats:italic> if <jats:italic>A<jats:sup>(n)<\/jats:sup><\/jats:italic> has the same degree as \u2298<jats:sup>(<jats:italic>n<\/jats:italic>+1)<\/jats:sup>. Simpson and Weitkamp [7] have shown that there is a high (high-1) incomplete <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline4\" \/> set in <jats:italic>L<\/jats:italic>. They have also shown that various other <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline4\" \/> sets are neither high nor low in <jats:italic>L<\/jats:italic>. Legrand [5] extended their results by showing that, if there is a real <jats:italic>x<\/jats:italic> such that <jats:italic>x<jats:sup>#<\/jats:sup><\/jats:italic> does not exist, then there is an element of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline3\" \/> that, for all <jats:italic>n<\/jats:italic>, is neither low-<jats:italic>n<\/jats:italic> nor high-<jats:italic>n<\/jats:italic>. In \u00a72, ZFC is used to show that, for all <jats:italic>n<\/jats:italic>, if <jats:italic>A<\/jats:italic> is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200031212_inline4\" \/> and low-<jats:italic>n<\/jats:italic> then <jats:italic>A<\/jats:italic> is Borel. The proof uses a strengthened version of Jensen's theorem on sequences of admissible ordinals that appears in [7, Simpson-Weitkamp].<\/jats:p>","DOI":"10.1017\/s0022481200031212","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T12:44:02Z","timestamp":1394714642000},"page":"352-359","source":"Crossref","is-referenced-by-count":0,"title":["On the Kleene degrees of <i>\u03a0<\/i><sub>1<\/sub><sup>1<\/sup> sets"],"prefix":"10.1017","volume":"51","author":[{"given":"Theodore A.","family":"Slaman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200031212_ref001","first-page":"5","volume-title":"Annales de la Facult\u00e9 des Sciences de l'Universit\u00e9 de Clermont-Ferrand","author":"Gandy","year":"1967"},{"key":"S0022481200031212_ref002","first-page":"685","volume":"43","author":"Harrington","year":"1978","journal-title":"Analytic determinacy and 0#"},{"key":"S0022481200031212_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/fm-108-2-83-88"},{"key":"S0022481200031212_ref011","unstructured":"Weitkamp G. , Kleene recursion over the continuum, Ph.D. thesis, Pennsylvania State University, University Park, Pennsylvania, 1980."},{"key":"S0022481200031212_ref003","unstructured":"Hrb\u00e1\u010dek K. , photocopied notes, 1981."},{"key":"S0022481200031212_ref004","unstructured":"Legrand S. , The Borel analogue of the Friedberg-Muchnik result (to appear)."},{"key":"S0022481200031212_ref005","unstructured":"Legrand S. , A type 2 version of a question of Sacks (to appear)."},{"key":"S0022481200031212_ref007","first-page":"356","volume":"48","author":"Simpson","year":"1983","journal-title":"High and low degrees of coanalytic sets"},{"key":"S0022481200031212_ref008","first-page":"374","volume":"36","author":"Solovay","year":"1971","journal-title":"Determinacy and type 2 recursion"},{"key":"S0022481200031212_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-48-3-313-320"},{"key":"S0022481200031212_ref006","first-page":"263","volume-title":"Proceedings of the sixth international congress on logic, methodology and philosophy of science, Hannover, 1979","author":"Simpson","year":"1982"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031212","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T05:53:59Z","timestamp":1683093239000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031212\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1986,6]]}},"alternative-id":["S0022481200031212"],"URL":"https:\/\/doi.org\/10.1017\/s0022481200031212","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,6]]}}}