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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-2-3,
author = {Lou van den Dries and Philip Ehrlich},
title = {Fields of surreal numbers and exponentiation},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {173-188},
zbl = {0974.03035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-2-3}
}
Lou van den Dries; Philip Ehrlich. Fields of surreal numbers and exponentiation. Fundamenta Mathematicae, Tome 167 (2001) pp. 173-188. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-2-3/