The surreal numbers as a universal H-field

Matthias Aschenbrenner, Lou Van Den Dries, Joris Van Der Hoeven

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1179-1199
Number of pages21
JournalJournal of the European Mathematical Society
Issue number4
StatePublished - 2019


  • Differential fields
  • Hardy fields
  • Surreal numbers
  • Transseries

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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