1Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A. 2Department of Philosophy Ohio University Athens, OH 45701, U.S.A.
Fundamenta Mathematicae, Tome 167 (2001) no. 2, pp. 173-188
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.
Keywords:
conways field surreal numbers its natural exponential function has elementary properties exponential field real numbers obtain ordinal bounds length products reciprocals exponentials logarithms surreal numbers terms lengths their inputs follows set surreal numbers length given ordinal subfield field surreal numbers only ordinal varepsilon number field even closed under surreal exponentiation elementary extension real exponential field
Affiliations des auteurs :
Lou van den Dries
1
;
Philip Ehrlich
2
1
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A.
2
Department of Philosophy Ohio University Athens, OH 45701, U.S.A.
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title = {Fields of surreal numbers and exponentiation},
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Lou van den Dries; Philip Ehrlich. Fields of surreal numbers and exponentiation. Fundamenta Mathematicae, Tome 167 (2001) no. 2, pp. 173-188. doi: 10.4064/fm167-2-3
