{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T12:08:19Z","timestamp":1709813299700},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14680,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1974,1]]},"abstract":"<jats:p>The purpose of this paper is to show that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum <jats:italic>\u03b2<\/jats:italic>-model for analysis). This is a much larger section of the second number class than the Church-Kleene constructive ordinals (although still only an initial segment of the <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200064410_inline1\" \/> ordinals). Arithmetic minimality means that if <jats:italic>H<\/jats:italic> is an \u201c<jats:italic>H<\/jats:italic>-set\u201d associated with an ordinal <jats:italic>\u03b1<\/jats:italic> in our system and <jats:italic>H\u2032<\/jats:italic> is an <jats:italic>H<\/jats:italic>-set associated with the same ordinal <jats:italic>\u03b1<\/jats:italic> in an <jats:italic>arbitrary<\/jats:italic> system of notations <jats:italic>S<\/jats:italic>, then <jats:italic>H<\/jats:italic> is arithmetical in <jats:italic>H\u2032<\/jats:italic>. Thus the arithmetical degrees associated with ordinals in our system are as low as possible.<\/jats:p><jats:p>In order to clarify the structure of degrees of unsolvability and, more generally, to gain a deeper insight into the power set of the integers, coarser but neater classifications than the structure of Turing degrees have been sought. Several hierarchies of sets of integers have been studied, each of which organizes a certain class of sets (or their degrees of unsolvability) into a well-ordering of levels with increasing complexity of nonrecursiveness appearing at each new level. The best known of these hierarchies is the Kleene hierarchy of arithmetical sets.<\/jats:p>","DOI":"10.2307\/2272637","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:29:41Z","timestamp":1146936581000},"page":"243-253","source":"Crossref","is-referenced-by-count":1,"title":["Systems of notations and the ramified analytical hierarchy"],"prefix":"10.1017","volume":"39","author":[{"given":"Joan D.","family":"Lukas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hilary","family":"Putnam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200064410_ref011","doi-asserted-by":"publisher","DOI":"10.2307\/1969604"},{"key":"S0022481200064410_ref001","first-page":"497","volume":"33","author":"Boolos","year":"1968","journal-title":"Degrees of unsolvability of constructible sets of integers"},{"key":"S0022481200064410_ref005","first-page":"429","volume":"35","author":"Enderton","year":"1970","journal-title":"A note on the hyperarithmetical hierarchy"},{"key":"S0022481200064410_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1964-0159755-9"},{"key":"S0022481200064410_ref003","first-page":"537","article-title":"A minimal model for set theory","volume":"3","author":"Cohen","year":"1964","journal-title":"Bulletin of the American Mathematical Society"},{"key":"S0022481200064410_ref010","first-page":"151","volume":"20","author":"Spector","year":"1955","journal-title":"Recursive well-orderings"},{"key":"S0022481200064410_ref008","unstructured":"Lukas J. D. , Systems of notations and the constructible-hierarchy, Doctoral Dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 1967."},{"key":"S0022481200064410_ref009","volume-title":"The theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200064410_ref006","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.25.4.220"},{"key":"S0022481200064410_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0184852-2"},{"key":"S0022481200064410_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1969-0242673-X"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200064410","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:20:43Z","timestamp":1559150443000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200064410\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,1]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1974,6]]}},"alternative-id":["S0022481200064410"],"URL":"https:\/\/doi.org\/10.2307\/2272637","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1974,1]]}}}