{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,2,10]],"date-time":"2023-02-10T10:55:51Z","timestamp":1676026551940},"reference-count":11,"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>Kruskal proved that finite trees are well-quasi-ordered by hom(e)omorphic embeddability. Friedman observed that this statement is not provable in predicative analysis. Friedman also proposed (see in [Simpson]) some stronger variants of the Kruskal theorem dealing with finite labeled trees under hom(e)omorphic embeddability with a certain gap-condition, where labels are arbitrary finite ordinals from a fixed initial segment of <jats:italic>\u03c9<\/jats:italic>. The corresponding limit statement, expressing that for all initial segments of <jats:italic>\u03c9<\/jats:italic> these labeled trees are well-quasi-ordered, is provable in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026499_inline1\" \/>-CA, but not in the analogous theory <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026499_inline1\" \/>-CA<jats:sub>0<\/jats:sub> with induction restricted to sets. Sch\u00fctte and Simpson proved that the one-dimensional case of Friedman's limit statement dealing with finite labeled intervals is not provable in Peano arithmetic. However, Friedman's gap-condition fails for finite trees labeled with transfinite ordinals. In [Gordeev 1] I proposed another gap-condition and proved the resulting one-dimensional modified statements for all (countable) transfinite ordinal-labels. The corresponding universal modified one-dimensional statement UM<jats:sup>1<\/jats:sup> is provable in (in fact, is equivalent to) the familiar theory ATR<jats:sub>0<\/jats:sub> whose proof-theoretic ordinal is \u0393<jats:sub>0<\/jats:sub>. In [Gordeev 1] I also announced that, in the general case of arbitrarily-branching finite trees labeled with transfinite ordinals, in the proof-theoretic sense the hierarchy of the limit modified statements M<jats:sub>&lt;\u03bb<\/jats:sub> (which are denoted by LM<jats:sub>\u03bb<\/jats:sub> in the present note) is as strong as the hierarchy of the familiar theories of iterated inductive definitions (more precisely, see [Gordeev 1, Concluding Remark 3]). In this note I present a \u201cpositive\u201d proof of the full universal modified statement UM, together with a short proof of the crucial \u201creverse\u201d results which is based on Okada's interpretation of the well-established ordinal notations of Buchholz corresponding to the theories of iterated inductive definitions. Formally the results are summarized in \u00a75 below.<\/jats:p>","DOI":"10.2307\/2274960","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:33:48Z","timestamp":1146954828000},"page":"157-181","source":"Crossref","is-referenced-by-count":7,"title":["Generalizations of the Kruskal-Friedman theorems"],"prefix":"10.1017","volume":"55","author":[{"given":"L.","family":"Gordeev","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026499_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70156-9"},{"key":"S0022481200026499_ref009","first-page":"557","volume":"52","author":"Okada","year":"1987","journal-title":"A simple relation between Buchholz's new system of ordinal notations and Takeuti's system of ordinal diagrams"},{"key":"S0022481200026499_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90052-7"},{"key":"S0022481200026499_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90078-9"},{"key":"S0022481200026499_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/1385-7258(77)90067-1"},{"key":"S0022481200026499_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0091894"},{"key":"S0022481200026499_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02007558"},{"key":"S0022481200026499_ref007","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-2.1.326"},{"key":"S0022481200026499_ref008","first-page":"210","article-title":"Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture","volume":"95","author":"Kruskal","year":"1960","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200026499_ref005","first-page":"100","volume":"54","author":"Gordeev","year":"1989","journal-title":"Generalizations of the one-dimensional version of the Kruskal-Friedman theorems"},{"key":"S0022481200026499_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90055-3"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026499","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:52:01Z","timestamp":1558216321000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026499\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026499"],"URL":"https:\/\/doi.org\/10.2307\/2274960","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}