{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T03:52:57Z","timestamp":1649130777948},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5671,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,9]]},"abstract":"Abstract<\/jats:title>Weiermann [18] introduces a new method to generate fast growing functions in order to get an elegant and perspicuous proof of a bounding theorem for provably total recursive functions in a formal theory, e.g., in PA<\/jats:italic>. His fast growing function \u03b8\u03b1\u03b7 is described as follows. For each ordinal a and natural number n<\/jats:italic> let denote a finitely branching, primitive recursive tree of ordinals, i.e., an ordinal as a label is attached to each node in the tree so that the labelling is compatible with the tree ordering. Then the tree is well founded and hence finite by K\u00f6nig's lemma. Define \u03b8\u03b1\u03b7n =the depth of the tree =the length of the longest branch in .<\/jats:p>We introduce new fast and slow growing functions in this mode of definitions and show that each of these majorizes provably total recursive functions in PA<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2586719","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:28Z","timestamp":1146938488000},"page":"897-925","source":"Crossref","is-referenced-by-count":3,"title":["Variations on a theme by Weiermann"],"prefix":"10.1017","volume":"63","author":[{"given":"Toshiyasu","family":"Arai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200014663_ref018","first-page":"52","volume":"61","author":"Weiermann","year":"1996","journal-title":"How to characterize provably total functions by local predicativity"},{"key":"S0022481200014663_ref003","unstructured":"Arai T. , Proof theory for theories of ordinals I.\u03a02-reflecting ordinals, manuscript, 02 1989."},{"key":"S0022481200014663_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(95)00029-G"},{"key":"S0022481200014663_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00003-L"},{"key":"S0022481200014663_ref021","unstructured":"Weiermann A. , Sometimes slow growing is fast growing, draft, 04 1996."},{"key":"S0022481200014663_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050089"},{"key":"S0022481200014663_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90078-9"},{"key":"S0022481200014663_ref017","unstructured":"Weiermann A. , Bounding derivation lengths with functions from the slow growing hierarchy, submitted."},{"key":"S0022481200014663_ref010","first-page":"171","volume-title":"Proof theory","author":"Cichon","year":"1992"},{"key":"S0022481200014663_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22156-3"},{"key":"S0022481200014663_ref020","unstructured":"Weiermann A. , Outer iteration together with lengths bounded transfinite recursion does not yield slow growingness, draft, 04 1996."},{"key":"S0022481200014663_ref014","first-page":"335","volume-title":"Logic Colloquium 78","author":"Schmerl","year":"1979"},{"key":"S0022481200014663_ref005","unstructured":"Arai T. , Systems of ordinal diagrams, manuscript, 08 1996."},{"key":"S0022481200014663_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00056-9"},{"key":"S0022481200014663_ref019","unstructured":"Weiermann A. , On the slow growing hierarchy via Cichon's fundamental sequences, manuscript, 02 1996."},{"key":"S0022481200014663_ref002","first-page":"322","article-title":"On Takeuti's fundamental conjecture (Japanese)","volume":"40","author":"Arai","year":"1988","journal-title":"S\u00fbgaku"},{"key":"S0022481200014663_ref013","first-page":"27","volume-title":"Proof theory","author":"Pohlers","year":"1992"},{"key":"S0022481200014663_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19940400212"},{"key":"S0022481200014663_ref016","volume-title":"Proof theory","author":"Takeuti","year":"1987"},{"key":"S0022481200014663_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(91)90027-J"},{"key":"S0022481200014663_ref001","unstructured":"Arai T. , Some results on cut-elimination, provable well-orderings, induction and reflection, to appear in Annals of Pure and Applied Logic ."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200014663","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T20:21:32Z","timestamp":1557519692000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200014663\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,9]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1998,9]]}},"alternative-id":["S0022481200014663"],"URL":"https:\/\/doi.org\/10.2307\/2586719","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,9]]}}}