{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T01:08:49Z","timestamp":1648861729999},"reference-count":12,"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":3663,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200008124_inline1\" \/>: two disjoint copies of a non-empty definable set <jats:italic>A<\/jats:italic> cannot be definably embedded into <jats:italic>A<\/jats:italic>, and principle CC of comparing cardinalities: for any two definable sets <jats:italic>A, B<\/jats:italic> either <jats:italic>A<\/jats:italic> definably embeds in <jats:italic>B<\/jats:italic> or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must satisfy <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200008124_inline1\" \/>.<\/jats:p><jats:p>Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200008124_inline2\" \/>: for no definable set <jats:italic>A<\/jats:italic> with more than one element can <jats:italic>A<\/jats:italic><jats:sup>2<\/jats:sup> definably embed into <jats:italic>A<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1080938837","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:29:26Z","timestamp":1109798966000},"page":"201-214","source":"Crossref","is-referenced-by-count":0,"title":["Approximate Euler characteristic, dimension, and weak pigeonhole principles"],"prefix":"10.1017","volume":"69","author":[{"given":"Jan","family":"Kraj\u00ed\u010dek","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200008124_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/421058"},{"key":"S0022481200008124_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0084232"},{"key":"S0022481200008124_ref012","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525919"},{"key":"S0022481200008124_ref005","doi-asserted-by":"publisher","DOI":"10.1112\/S0024611500012557"},{"key":"S0022481200008124_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/1970573"},{"key":"S0022481200008124_ref004","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511529948"},{"key":"S0022481200008124_ref001","first-page":"1","volume-title":"Feasible mathematics II","author":"Ajtai","year":"1994"},{"key":"S0022481200008124_ref006","volume-title":"Archive for Mathematical Logic","author":"Kraj\u00ed\u010dek","year":"2001"},{"key":"S0022481200008124_ref003","first-page":"107","article-title":"Definable sets over finite fields","volume":"427","author":"Chatzidakis","year":"1992","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"key":"S0022481200008124_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF01200117"},{"key":"S0022481200008124_ref009","first-page":"1235","volume":"53","author":"Paris","year":"1988","journal-title":"Provability of the pigeonhole principle and the existence of infinitely many primes"},{"key":"S0022481200008124_ref007","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240070103"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200008124","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T21:28:53Z","timestamp":1557178133000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200008124\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,3]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2004,3]]}},"alternative-id":["S0022481200008124"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1080938837","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,3]]}}}