{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T06:10:59Z","timestamp":1683094259847},"reference-count":11,"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":9507,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (<jats:italic>P<\/jats:italic><jats:sup>\u2212<\/jats:sup>) together with <jats:italic>\u03a3<jats:sub>2<\/jats:sub><\/jats:italic>-collection (<jats:italic>B<\/jats:italic>\u03a3<jats:sub>2<\/jats:sub>). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each <jats:italic>n<\/jats:italic>, the existence of an incomplete recursively enumerable set that is neither <jats:italic>low<jats:sub>n<\/jats:sub><\/jats:italic> nor <jats:italic>high<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>-1<\/jats:sub>, while true, cannot be established in <jats:italic>P<jats:sup>\u2212<\/jats:sup> + B\u03a3<jats:sub>n+1<\/jats:sub><\/jats:italic>. Consequently, no bounded fragment of first order arithmetic establishes the facts that the <jats:italic>high<jats:sub>n<\/jats:sub><\/jats:italic> and <jats:italic>low<jats:sub>n<\/jats:sub><\/jats:italic> jump hierarchies are proper on the recursively enumerable degrees.<\/jats:p>","DOI":"10.1017\/s0022481200029042","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T12:41:05Z","timestamp":1394714465000},"page":"212-221","source":"Crossref","is-referenced-by-count":1,"title":["<i>\u03a3<sub>2<\/sub><\/i>-collection and the infinite injury priority method"],"prefix":"10.1017","volume":"53","author":[{"given":"Michael E.","family":"Mytilinaios","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":"S0022481200029042_ref011","volume-title":"Recursively enumerable sets and degrees","author":"Soare","year":"1986"},{"key":"S0022481200029042_ref009","first-page":"351","article-title":"On the jump of an \u03b1-recursively enumerable set","volume":"217","author":"Shore","year":"1976","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200029042_ref004","unstructured":"Kirby L. A. S. , Initial segments of models of arithmetic, Ph.D. thesis, University of Manchester, Manchester, 1977."},{"key":"S0022481200029042_ref002","unstructured":"Groszek M. J. and Slaman T. A. , Foundations of the priority method. I: Finite and infinite injury (to appear)."},{"key":"S0022481200029042_ref008","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-2-171-179"},{"key":"S0022481200029042_ref010","unstructured":"Slaman T. A. and Woodin W. H. , \u03a31-collection and the finite injury priority method (to appear)."},{"key":"S0022481200029042_ref005","first-page":"199","volume-title":"Logic Colloquium \u201877","author":"Kirby","year":"1978"},{"key":"S0022481200029042_ref001","unstructured":"Groszek M. J. and Mytilinaios M. E. , \u03a32 -induction and the construction of a high degree(to appear)."},{"key":"S0022481200029042_ref003","first-page":"599","article-title":"Pseudo jump operators. I: The r. e. case","volume":"275","author":"Jockusch","year":"1983","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200029042_ref006","unstructured":"Mytilinaios M. E. , Finite injury and \u03a32-induction (to appear)."},{"key":"S0022481200029042_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1963-0155747-3"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200029042","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T05:48:48Z","timestamp":1683092928000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200029042\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1988,3]]}},"alternative-id":["S0022481200029042"],"URL":"https:\/\/doi.org\/10.1017\/s0022481200029042","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,3]]}}}