{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T09:20:35Z","timestamp":1775467235057,"version":"3.50.1"},"reference-count":22,"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":15717,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1971,3]]},"abstract":"<jats:p>In this paper we describe generalizations of several approaches to the hyperarithmetic hierarchy, show how they are related to the Kripke-Platek theory of admissible ordinals and sets, and study conditions under which the various approaches remain equivalent.<\/jats:p><jats:p>To put matters in some perspective, let us first review various approaches to the theory of hyperarithmetic sets. For most purposes, it is convenient to first define the <jats:italic>semi-hyperarithmetic<\/jats:italic> (semi-HA) subsets of <jats:italic>N<\/jats:italic>. A set is then said to be <jats:italic>hyperarithmetic<\/jats:italic> (HA) if both it and its complement are semi-HA. A total number-theoretic function is HA if its graph is HA.<\/jats:p>","DOI":"10.2307\/2271519","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:05:56Z","timestamp":1146949556000},"page":"108-120","source":"Crossref","is-referenced-by-count":49,"title":["The next admissible set"],"prefix":"10.1017","volume":"36","author":[{"given":"K. J.","family":"Barwise","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R. O.","family":"Gandy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Y. N.","family":"Moschovakis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200083973_ref021","unstructured":"Platek R. , Foundations of recursion theory, Ph.D. Thesis and supplement, Stanford Univ. 1966."},{"key":"S0022481200083973_ref020","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-70-03744-0"},{"key":"S0022481200083973_ref019","unstructured":"Moschovakis Y. N. , Abstract computability and invariant definability, this Journal (to appear) (referred to as ACID)."},{"key":"S0022481200083973_ref015","first-page":"162","article-title":"Transfinite recursion on admissible ordinals, I, II (abstract)","volume":"29","author":"Kripke","year":"1964","journal-title":"this Journal"},{"key":"S0022481200083973_ref013","first-page":"321","article-title":"A survey of proof theory","volume":"33","author":"Kreisel","year":"1968","journal-title":"this Journal"},{"key":"S0022481200083973_ref010","first-page":"1","article-title":"Recursive functionals and quantifiers of finite types, I","volume":"91","author":"Kleene","year":"1959","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200083973_ref006","first-page":"188","article-title":"The classical and the \u03c9-complete arithmetic","volume":"23","author":"Grzegorczyk","year":"1958","journal-title":"this Journal"},{"key":"S0022481200083973_ref014","first-page":"318","article-title":"Metarecursive sets","volume":"30","author":"Kreisel","year":"1965","journal-title":"this Journal"},{"key":"S0022481200083973_ref003","unstructured":"Barwise J. , The Gandy-Kreisel-Tait theorem for countable sets, (in preparation)."},{"key":"S0022481200083973_ref022","first-page":"97","volume-title":"Infinitistic Methods","author":"Spector","year":"1961"},{"key":"S0022481200083973_ref018","first-page":"427","article-title":"Abstract first order computability, I, II","volume":"138","author":"Moschovakis","year":"1969","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200083973_ref017","first-page":"37","article-title":"Constructive definitions of certain analytic sets of numbers","volume":"24","author":"Lorenzen","year":"1959","journal-title":"this Journal"},{"key":"S0022481200083973_ref002","first-page":"409","article-title":"Applications of strict-\u03a011 predicates to infinitary logic","volume":"34","author":"Barwise","year":"1969","journal-title":"this Journal"},{"key":"S0022481200083973_ref009","first-page":"23","article-title":"Quantification of number theoretic functions","volume":"14","author":"Kleene","year":"1959","journal-title":"Compositio Mathematica"},{"key":"S0022481200083973_ref016","first-page":"446","article-title":"Implicit definability and infinitary languages","volume":"33","author":"Kunen","year":"1968","journal-title":"this Journal"},{"key":"S0022481200083973_ref001","first-page":"226","article-title":"Infinitary logic and admissible sets","volume":"34","author":"Barwise","year":"1969","journal-title":"this Journal"},{"key":"S0022481200083973_ref005","first-page":"577","article-title":"Set existence","volume":"8","author":"Gandy","year":"1960","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences"},{"key":"S0022481200083973_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71509-X"},{"key":"S0022481200083973_ref007","volume-title":"Proceedings of the Summer Institute on Axiomatic Set Theory","author":"Jensen","year":"1967"},{"key":"S0022481200083973_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1955-09896-3"},{"key":"S0022481200083973_ref011","first-page":"103","volume-title":"Infinitistic Methods","author":"Kreisel","year":"1961"},{"key":"S0022481200083973_ref012","first-page":"190","volume-title":"The Theory of Models","author":"Kreisel","year":"1965"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200083973","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T20:20:24Z","timestamp":1559334024000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200083973\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1971,3]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1971,3]]}},"alternative-id":["S0022481200083973"],"URL":"https:\/\/doi.org\/10.2307\/2271519","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1971,3]]}}}