The surreal numbers as a universal $H$-field

Matthias Aschenbrenner; Lou van den Dries; Joris van der Hoeven

doi:10.4171/jems/858

JournalsjemsVol. 21, No. 4pp. 1179–1199

The surreal numbers as a universal $H$ -field

Matthias Aschenbrenner
University of California Los Angeles, USA
Lou van den Dries
University of Illinois at Urbana-Champaign, Urbana, USA
Joris van der Hoeven
Ecole Polytechnique, Palaiseau, France

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Abstract

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.

Cite this article

Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven, The surreal numbers as a universal $H$ -field. J. Eur. Math. Soc. 21 (2019), no. 4, pp. 1179–1199

DOI 10.4171/JEMS/858