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For both logics we obtain as an upper bound for this number an exponential tower of height linear in <jats:italic>n<\/jats:italic>. We prove this by counting types with respect to a fixed quantifier rank. As lower bounds we obtain for both FO and MSO an exponential tower of height in the order of a rational power of <jats:italic>n<\/jats:italic>. We show these lower bounds by giving concrete formulas defining word representations of levels of the cumulative hierarchy of sets. In addition, we consider the L\u00f6wenheim-Skolem and Hanf numbers of these logics on words and obtain similar bounds for these as well.<\/jats:p>","DOI":"10.1007\/978-3-031-08740-0_11","type":"book-chapter","created":{"date-parts":[[2022,6,25]],"date-time":"2022-06-25T14:33:51Z","timestamp":1656167631000},"page":"125-138","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Defining Long Words Succinctly in\u00a0FO and\u00a0MSO"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9117-8124","authenticated-orcid":false,"given":"Lauri","family":"Hella","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7301-939X","authenticated-orcid":false,"given":"Miikka","family":"Vilander","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,6,26]]},"reference":[{"key":"11_CR1","doi-asserted-by":"crossref","unstructured":"Ebbinghaus, H.D.: L\u00f6wenheim-Skolem theorems. In: Gabbay, D., Thagard, P., Woods, J., Jacquette, D. (eds.) Philosophy of Logic, Handbook of the Philosophy of Science. Elsevier Science (2006)","DOI":"10.1016\/B978-044451541-4\/50018-X"},{"key":"11_CR2","doi-asserted-by":"crossref","unstructured":"Ebbinghaus, H., Flum, J.: Finite model theory. Perspectives in Mathematical Logic, Springer, Germany (1995)","DOI":"10.1007\/978-3-662-03182-7"},{"key":"11_CR3","doi-asserted-by":"publisher","unstructured":"Ellul, K., Krawetz, B., Shallit, J., Wang, M.W.: Regular expressions: new results and open problems. J. Autom. Lang. Comb. 10(4), 407\u2013437 (2005). https:\/\/doi.org\/10.25596\/jalc-2005-407","DOI":"10.25596\/jalc-2005-407"},{"key":"11_CR4","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1002\/malq.19960420145","volume":"42","author":"M Grohe","year":"1996","unstructured":"Grohe, M.: Some remarks on finite L\u00f6wenheim-Skolem theorems. Math. Log. 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