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We show that the natural embedding of the differential field of transseries into Conway’s field of surreal numbers with the Berarducci–Mantova derivation is an elementary embedding. We also prove that any Hardy field embeds into the field of surreals with the Berarducci–Mantova derivation.
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Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven, The surreal numbers as a universal -field. J. Eur. Math. Soc. 21 (2019), no. 4, pp. 1179–1199DOI 10.4171/JEMS/858