{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T15:04:20Z","timestamp":1648911860318},"reference-count":15,"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":8777,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:p>The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)<\/jats:p><jats:p>As go the integers so go the ordinals. One motivation of <jats:italic>\u03b1<\/jats:italic>-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of <jats:italic>\u03c9<\/jats:italic>, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of <jats:italic>\u03b1<\/jats:italic>-recursion theory.<\/jats:p><jats:p>When working in the theory KP \u2013 Foundation Schema (hereinafter called KP<jats:sup>\u2212<\/jats:sup>), one should really not call it <jats:italic>\u03b1<\/jats:italic>-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name <jats:italic>\u03b2<\/jats:italic>-recursion theory refers to what would be <jats:italic>\u03b1<\/jats:italic>-recursion theory only it includes also inadmissible ordinals, we call the subject of study here <jats:italic>\u03b3<\/jats:italic>-recursion theory. This answers a question by Sacks and S. Friedman, \u201cWhat is <jats:italic>\u03b3<\/jats:italic>-recursion theory?\u201d<\/jats:p>","DOI":"10.2307\/2274962","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:33:48Z","timestamp":1146940428000},"page":"194-206","source":"Crossref","is-referenced-by-count":1,"title":["An introduction to <i>\u03b3<\/i>-recursion theory (or what to do in KP \u2013 Foundation)"],"prefix":"10.1017","volume":"55","author":[{"given":"Robert S.","family":"Lubarsky","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026517_ref013","first-page":"1","article-title":"Post's problem, admissible ordinals, and regularity","volume":"124","author":"Sacks","year":"1966","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200026517_ref012","first-page":"212","volume":"53","author":"Mytilinaios","year":"1988","journal-title":"\u03a32 collection and the infinite injury priority method"},{"key":"S0022481200026517_ref008","volume-title":"Annals of Pure and Applied Logic","author":"Lubarsky"},{"key":"S0022481200026517_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0059290"},{"key":"S0022481200026517_ref015","unstructured":"Slaman T. A. and Woodin W. H. , \u03a31 collection and the finite injury priority method (to appear)."},{"key":"S0022481200026517_ref002","unstructured":"Chong C. T. , Maximal sets and fragments of Peano arithmetic (to appear)."},{"key":"S0022481200026517_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200026517_ref005","first-page":"173","article-title":"\u03b2-recursion theory","volume":"255","author":"Friedman","year":"1979","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200026517_ref009","unstructured":"Lubarsky R. S. , Correction to [L3], this Journal, vol. 53 (1988), pp. 103\u2013104."},{"key":"S0022481200026517_ref004","first-page":"389","volume":"33","author":"Driscoll","year":"1968","journal-title":"Metarecursively enumerable sets and their metadegrees"},{"key":"S0022481200026517_ref007","unstructured":"Groszekand M. Slaman T. A. , Foundations of the priority method. I: Finite and infinite injury (to appear)."},{"key":"S0022481200026517_ref010","first-page":"208","volume":"52","author":"Lubarsky","year":"1987","journal-title":"Simple r.e. degree structures"},{"key":"S0022481200026517_ref011","first-page":"38","volume":"54","author":"Mytilinaios","year":"1989","journal-title":"Finite injury and \u03a32 induction"},{"key":"S0022481200026517_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(80)90042-0"},{"key":"S0022481200026517_ref014","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"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026517","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T17:51:30Z","timestamp":1558201890000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026517\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026517"],"URL":"https:\/\/doi.org\/10.2307\/2274962","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}