Fields of surreal numbers and exponentiation

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. Introduction. Conway [1] introduced the ordered field No of surreal numbers, which extends the field R of real numbers. (See Section 1 for a brief account of No.) Gonshor ([6], Ch. 10) followed suggestions by Kruskal and defined an exponential function exp : No→ No such that exp(x) = e for x ∈ R. In Section 2 below we show that No with exp is an elementary extension of the real exponential field: elementary statements true in the real exponential field remain true in the exponential field of surreal numbers. (See Wilkie [10], and Macintyre and Wilkie [9] for information on the elementary theory of the real exponential field.) This result relating real and surreal exponentiation was also noticed by A. Macintyre, by M. H. Mourgues, and by J. Lurie [8], and answers a question of the first author in [2], p. 8. The proof below consists in equipping No with even further structure, by extending the restricted analytic functions from the real field to No, and verifying that the axioms in [3] for the model-complete theory Tan,exp are satisfied by the thus expanded No. The original content of the paper lies almost entirely in Sections 3–5, which contain the following results. Let No(λ) be the set of surreals of 2000 Mathematics Subject Classification: Primary 03C64, 03C65, 03H05, 12J15, 20F60; Secondary 04A10, 06F.