{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T04:56:23Z","timestamp":1768971383159,"version":"3.49.0"},"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":9142,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1989,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Working in the language of first-order arithmetic we consider models of the base theory <jats:italic>P<\/jats:italic><jats:sup>\u2212<\/jats:sup>. Suppose <jats:italic>M<\/jats:italic> is a model of <jats:italic>P<\/jats:italic><jats:sup>\u2212<\/jats:sup> and let <jats:italic>M<\/jats:italic> satisfy induction for <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub>-formulas. First it is shown that the Friedberg-Muchnik finite injury argument can be performed inside <jats:italic>M<\/jats:italic>, and then, using a blocking method for the requirements, we prove that the Sacks splitting construction can be done in <jats:italic>M<\/jats:italic>. So, the \u201camount\u201d of induction needed to perform the known finite injury priority arguments is <jats:italic>\u03a3<\/jats:italic><jats:sub>1<\/jats:sub>-induction.<\/jats:p>","DOI":"10.2307\/2275013","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:29:26Z","timestamp":1146954566000},"page":"38-49","source":"Crossref","is-referenced-by-count":25,"title":["Finite injury and <i>\u03a3<\/i><sub>1<\/sub>-induction"],"prefix":"10.1017","volume":"54","author":[{"given":"Michael","family":"Mytilinaios","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200027560_ref011","unstructured":"Simpson S. G. , private correspondence, 05 24, 1984. (Contained the proof of Theorem 3.2 and a discussion of the program.)"},{"key":"S0022481200027560_ref002","unstructured":"Kirby L. A. S. , Initial segments of models of arithmetic, Ph.D. thesis, University of Manchester, Manchester, 1977."},{"key":"S0022481200027560_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/1970214"},{"key":"S0022481200027560_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90004-6"},{"key":"S0022481200027560_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71230-8"},{"key":"S0022481200027560_ref007","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200027560_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1944-08111-1"},{"key":"S0022481200027560_ref010","first-page":"65","article-title":"Splitting an \u03b1-recursively enumerable set","volume":"204","author":"Shore","year":"1975","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200027560_ref001","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.43.2.236"},{"key":"S0022481200027560_ref004","first-page":"194","article-title":"On the unsolvability of the problem of reducibility in the theory of algorithms","volume":"108","author":"Muchnik","year":"1956","journal-title":"Doklady Akademii Nauk SSSR"},{"key":"S0022481200027560_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)72003-2"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200027560","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T21:01:52Z","timestamp":1558299712000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200027560\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1989,3]]}},"alternative-id":["S0022481200027560"],"URL":"https:\/\/doi.org\/10.2307\/2275013","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,3]]}}}