Fields of surreal numbers and exponentiation

Lou van den Dries, and Philip Ehrlich. "Fields of surreal numbers and exponentiation." Fundamenta Mathematicae 167.2 (2001): 173-188. <http://eudml.org/doc/282499>.

@article{LouvandenDries2001,
abstract = {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.},
author = {Lou van den Dries, Philip Ehrlich},
journal = {Fundamenta Mathematicae},
keywords = {surreal number; exponential field},
language = {eng},
number = {2},
pages = {173-188},
title = {Fields of surreal numbers and exponentiation},
url = {http://eudml.org/doc/282499},
volume = {167},
year = {2001},
}

TY - JOUR
AU - Lou van den Dries
AU - Philip Ehrlich
TI - Fields of surreal numbers and exponentiation
JO - Fundamenta Mathematicae
PY - 2001
VL - 167
IS - 2
SP - 173
EP - 188
AB - 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.
LA - eng
KW - surreal number; exponential field
UR - http://eudml.org/doc/282499
ER -