{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T04:37:37Z","timestamp":1771562257451,"version":"3.50.1"},"reference-count":5,"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:title>Abstract<\/jats:title><jats:p>We study <jats:italic>increasing F-sequences<\/jats:italic>, where <jats:italic>F<\/jats:italic> is a dilator: an increasing <jats:italic>F<\/jats:italic>-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step <jats:italic>x<\/jats:italic> where <jats:italic>F(x)<\/jats:italic> is reached (at every step <jats:italic>x<\/jats:italic> + 1 we use the same process as in <jats:italic>decreasing F-sequences<\/jats:italic>, cf. [2], but with \u201c+ 1\u201d instead of \u201c\u22121\u201d). By induction on dilators, we shall prove that every increasing <jats:italic>F<\/jats:italic>-sequence terminates and moreover we can determine for every dilator <jats:italic>F<\/jats:italic> the point where the increasing <jats:italic>F<\/jats:italic>-sequence terminates.<\/jats:p><jats:p>We apply these results to <jats:italic>inverse Goodstein sequences<\/jats:italic>, i.e. increasing (1 + Id)<jats:sub>(<jats:italic>\u03c9<\/jats:italic>)<\/jats:sub>-sequences. We show that the theorem<\/jats:p><jats:p><jats:disp-quote><jats:p><jats:italic>every inverse Goodstein sequence terminates<\/jats:italic><\/jats:p><\/jats:disp-quote><\/jats:p><jats:p>(a combinatorial theorem about ordinal numbers) is not provable in ID<jats:sub>1<\/jats:sub>.<\/jats:p><jats:p>For a general presentation of the results stated in this paper, see [1].<\/jats:p><jats:p>We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25\u201331].<\/jats:p>","DOI":"10.2307\/2274952","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:33:48Z","timestamp":1146940428000},"page":"32-40","source":"Crossref","is-referenced-by-count":2,"title":["Some uses of dilators in combinatorial problems. II"],"prefix":"10.1017","volume":"55","author":[{"given":"V. Michele","family":"Abrusci","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Yves","family":"Girard","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jacques","family":"van de Wiele","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026414_ref003","first-page":"35","volume":"37","author":"Aczel","year":"1972","journal-title":"Describing ordinals using functionals of transfinite type"},{"key":"S0022481200026414_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/065\/891241"},{"key":"S0022481200026414_ref005","volume-title":"Ptykes in G\u00f6dels T und verallgemeinerte Rekursion \u00fcber Mengen und Ordinalzahlen","author":"P\u00e4ppinghaus","year":"1985"},{"key":"S0022481200026414_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/065\/891240"},{"key":"S0022481200026414_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(81)90016-4"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026414","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T17:52:42Z","timestamp":1558201962000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026414\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026414"],"URL":"https:\/\/doi.org\/10.2307\/2274952","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}