Fields of surreal numbers and exponentiation

Volume 167 / 2001

Lou van den Dries, Philip Ehrlich Fundamenta Mathematicae 167 (2001), 173-188 MSC: Primary 03C64, 03C65, 03H05, 12J15, 20F60; Secondary 04A10, 06F. DOI: 10.4064/fm167-2-3


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 $\varepsilon $-number. In that case, this field is even closed under surreal exponentiation, and is an elementary extension of the real exponential field.


  • Lou van den DriesDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    Urbana, IL 61801, U.S.A.
  • Philip EhrlichDepartment of Philosophy
    Ohio University
    Athens, OH 45701, U.S.A.

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