{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T16:09:47Z","timestamp":1762272587112},"reference-count":16,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2021,4]]},"abstract":"<jats:p> Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by establishing a link between said rank and the [Formula: see text]-Scott height of the [Formula: see text]-sized models of [Formula: see text], and yields to the following descriptive set-theoretical analog of Shelah\u2019s Main Gap Theorem: Given a countable complete first-order theory [Formula: see text], either [Formula: see text] is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is [Formula: see text]), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah\u2019s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of [Formula: see text], and provide a characterization of categoricity of [Formula: see text] in terms of the descriptive set-theoretical complexity of [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0219061320500257","type":"journal-article","created":{"date-parts":[[2020,6,24]],"date-time":"2020-06-24T16:06:01Z","timestamp":1593014761000},"page":"2050025","source":"Crossref","is-referenced-by-count":2,"title":["A descriptive Main Gap Theorem"],"prefix":"10.1142","volume":"21","author":[{"given":"Francesco","family":"Mangraviti","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Philosophie I, Ruhr Universit\u00e4t Bochum, Universit\u00e4tsstr. 150, 44801 Bochum, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luca","family":"Motto Ros","sequence":"additional","affiliation":[{"name":"Dipartimento di matematica \u00abGiuseppe Peano\u00bb, Universit\u00e0 di Torino, Via Carlo Alberto 10, 10123 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2020,6,24]]},"reference":[{"key":"S0219061320500257BIB002","series-title":"Perspectives in Mathematical Logic","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-07330-8","volume-title":"Fundamentals of Stability Theory","author":"Baldwin J. 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