Geodesic
Fields of surreal numbers and exponentiation
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.
Fundamenta Mathematicae, Tome 167 (2001) no. 2, pp. 173-188 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/fm167-2-3
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

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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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

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