{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T17:38:26Z","timestamp":1775497106048,"version":"3.50.1"},"reference-count":13,"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":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"Let A<\/jats:italic> be a subset of \u03c9, the set of natural numbers. The degree<\/jats:italic> of A<\/jats:italic> is its degree of recursive unsolvability. We say that A<\/jats:italic> is rich<\/jats:italic> if every degree above that of A<\/jats:italic> is represented by a subset of A<\/jats:italic>. We say that A<\/jats:italic> is poor<\/jats:italic> if no degree strictly above that of A<\/jats:italic> is represented by a subset of A<\/jats:italic>. The existence of infinite poor (and hence nonrich) sets was proved by Soare [9].<\/jats:p>Theorem<\/jats:sc> 1. Suppose that A is infinite and not rich. Then every hyperarith-metical subset H of \u03c9 is recursive in A<\/jats:italic>.<\/jats:p>In the special case when H<\/jats:italic> is arithmetical, Theorem 1 was proved by Jockusch [4] who employed a degree-theoretic analysis of Ramsey's theorem [3]. In our proof of Theorem 1 we employ a similar, degree-theoretic analysis of a certain generalization of Ramsey's theorem. The generalization of Ramsey's theorem is due to Nash-Williams [6]. If A<\/jats:italic> \u2286 \u03c9 we write [A<\/jats:italic>]\u03c9<\/jats:sup> for the set of all infinite subsets of A<\/jats:italic>. If P<\/jats:italic> \u2286 [\u03c9]\u03c9<\/jats:sup> we let H(P)<\/jats:italic> be the set of all infinite sets A<\/jats:italic> such that either [A<\/jats:italic>]\u03c9<\/jats:sup> \u2286 P<\/jats:italic> = \u2205. Nash-Williams' theorem is essentially the statement that if P<\/jats:italic> \u2286 [\u03c9]\u03c9<\/jats:sup> is clopen<\/jats:italic> (in the usual, Baire topology on [\u03c9]\u03c9<\/jats:sup>) then H(P)<\/jats:italic> is nonempty. Subsequent, further generalizations of Ramsey's theorem were proved by Galvin and Prikry [1], Silver [8], Mathias [5], and analyzed degree-theoretically by Solovay [10]; those results are not needed for this paper.<\/jats:p>","DOI":"10.2307\/2271956","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:46:34Z","timestamp":1146937594000},"page":"135-138","source":"Crossref","is-referenced-by-count":14,"title":["Sets which do not have subsets of every higher degree"],"prefix":"10.1017","volume":"43","author":[{"given":"Stephen G.","family":"Simpson","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049963_ref012","first-page":"235","article-title":"Some systems of second order arithmetic and their use","volume":"1","author":"Friedman","journal-title":"Proceedings of the International Congress of Mathematicians, Vancouver, 1974"},{"key":"S0022481200049963_ref010","volume-title":"Transactions of the American Mathematical Society","author":"Solovay","year":"1975"},{"key":"S0022481200049963_ref007","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200049963_ref006","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100038603"},{"key":"S0022481200049963_ref005","volume-title":"Annals of Mathematical Logic","author":"Mathias"},{"key":"S0022481200049963_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02787575"},{"key":"S0022481200049963_ref001","first-page":"193","volume":"38","author":"Galvin","year":"1973","journal-title":"Borel sets and Ramsey's theorem"},{"key":"S0022481200049963_ref003","first-page":"268","volume":"37","author":"Jockusch","year":"1972","journal-title":"Ramsey's theorem and recursion theory"},{"key":"S0022481200049963_ref008","first-page":"60","volume":"35","author":"Silver","year":"1970","journal-title":"Every analytic set is Ramsey"},{"key":"S0022481200049963_ref002","first-page":"81","volume":"37","author":"Grilliot","year":"1972","journal-title":"Omitting types: applications to recursion theory"},{"key":"S0022481200049963_ref011","first-page":"151","volume":"20","author":"Spector","year":"1955","journal-title":"Recursive well-orderings"},{"key":"S0022481200049963_ref009","first-page":"53","volume":"34","author":"Soare","year":"1969","journal-title":"Sets with no subset of higher degree"},{"key":"S0022481200049963_ref013","unstructured":"Steel J. , Determinateness and subsystems of analysis, Ph.D. Thesis, University of California at Berkeley, 1977."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049963","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T16:19:45Z","timestamp":1558973985000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049963\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,3]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1978,3]]}},"alternative-id":["S0022481200049963"],"URL":"https:\/\/doi.org\/10.2307\/2271956","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,3]]}}}