Fields of surreal numbers and exponentiation

Lou Van Den Dries, Philip Ehrlich

Research output: Contribution to journalArticlepeer-review


We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case, this field is even closed under surreal exponentiation, and is an elementary extension of the real exponential field.

Original languageEnglish (US)
Pages (from-to)173-188
Number of pages16
JournalFundamenta Mathematicae
Issue number2
StatePublished - 2001


  • Exponential fields
  • Surreal numbers

ASJC Scopus subject areas

  • Algebra and Number Theory

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