{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T08:50:41Z","timestamp":1772441441538,"version":"3.50.1"},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7681,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1993,3]]},"abstract":"<jats:p>A first-order sentence <jats:italic>\u03a6<\/jats:italic> is <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub> if there is a quantifier-free formula <jats:italic>\u0398<\/jats:italic> such that <jats:italic>\u03a6<\/jats:italic> has the form <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline1\"\/>. The <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub>-<jats:italic>theory<\/jats:italic> of a structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline2\"\/> for a language \u2112  is the set of <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub>-sentences of <jats:italic>\u2112<\/jats:italic> true in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline2\"\/>. It was shown independently by Lerman and Shore (see [Le, Theorem VII.4.4]) that the <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub>-theory of the structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline3\"\/> = \u3008<jats:italic>D<\/jats:italic>, \u2264 \u3009 is decidable, where <jats:italic>D<\/jats:italic> is the set of degrees of unsolvability and \u2264 is the standard ordering of <jats:italic>D<\/jats:italic>. This result is optimal in the sense that the <jats:italic>\u03a3<\/jats:italic><jats:sub>3<\/jats:sub>-theory of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline3\"\/> is undecidable, a result due to J. Schmerl. (For a proof, see [Le, Theorem VII.4.5]. As Lerman has pointed out, this proof should be corrected by defining <jats:italic>\u03b8<jats:sub>\u03c3<\/jats:sub><\/jats:italic> to be \u2200<jats:italic>x\u03c3<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>x<\/jats:italic>) rather than \u2200<jats:italic>x(\u03c8(x)<\/jats:italic>\u2192 <jats:italic>\u03c3<jats:sub>1<\/jats:sub>(x))<\/jats:italic>.) Nonetheless, in this paper we extend the decidability result of Lerman and Shore by showing that the <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub>-theory of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline3\"\/> is decidable, where \u22c3 is the least upper bound operator and <jats:bold>0<\/jats:bold> is the least degree. Of course \u22c3 is definable in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline3\"\/>, but many interesting degree-theoretic results are expressible as <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub>-sentences in the language of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline3\"\/><jats:sub>\u222a<\/jats:sub> but not as <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub>-sentences in the language of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline3\"\/>. For instance, Simpson observed that the Posner-Robinson cupping theorem could be used to show that for any nonzero degrees <jats:bold>a<\/jats:bold>, <jats:bold>b<\/jats:bold>, there is a degree g such that <jats:bold>b<\/jats:bold> \u2264 <jats:bold>a<\/jats:bold> \u22c3 <jats:bold>g<\/jats:bold>, and <jats:bold>b<\/jats:bold> \u22e0 <jats:bold>g<\/jats:bold> (see [PR, Corollary 6]). However, the Posner-Robinson technique does not seem to suffice to decide the <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub>-theory of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002171X_inline3\"\/><jats:sub>\u222a<\/jats:sub>. We introduce instead a new method for coding a set into the join of two other sets and use it to decide this theory.<\/jats:p>","DOI":"10.2307\/2275332","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:47:31Z","timestamp":1146955651000},"page":"193-204","source":"Crossref","is-referenced-by-count":8,"title":["On the <i>\u03a3<\/i><sub>2<\/sub>-theory of the upper semilattice of Turing degrees"],"prefix":"10.1017","volume":"58","author":[{"suffix":"Jr","given":"Carl G.","family":"Jockusch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Theodore A.","family":"Slaman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120002171X_ref004","first-page":"705","volume":"46","year":"1981","journal-title":"Degrees joining to 0\u2032"},{"key":"S002248120002171X_ref001","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-68-03513-8"},{"key":"S002248120002171X_ref002","volume-title":"Perspectives in Mathematical Logic","year":"1983"},{"key":"S002248120002171X_ref003","first-page":"1","volume-title":"Transactions of the American Mathematical Society","volume":"257","year":"1980"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120002171X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T19:14:33Z","timestamp":1558034073000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120002171X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1993,3]]}},"alternative-id":["S002248120002171X"],"URL":"https:\/\/doi.org\/10.2307\/2275332","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,3]]}}}