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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Conway's field of surreal numbers


Author: Norman L. Alling
Journal: Trans. Amer. Math. Soc. 287 (1985), 365-386
MSC: Primary 04A10; Secondary 06A05, 12J15, 12J25
MathSciNet review: 766225
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Abstract: Conway introduced the Field $ {\mathbf{No}}$ of numbers, which Knuth has called the surreal numbers. $ {\mathbf{No}}$ is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff's $ {\eta _\xi }$ condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of $ {\mathbf{No}}$. In the process, a tower of subfields, $ \xi {\mathbf{No}}$, is defined, each of which is a real-closed subfield of $ {\mathbf{No}}$ that is an $ {\eta _\xi }$-set. These fields all have Conway partitions. This structure allows the author to prove that every pseudo-convergent sequence in $ {\mathbf{No}}$ has a unique limit in $ {\mathbf{No}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0766225-7
PII: S 0002-9947(1985)0766225-7
Keywords: Surreal numbers, real-closed fields, valuation theory, pseudo-convergent sequences, $ {\eta _\xi }$-sets, fields of formal power series
Article copyright: © Copyright 1985 American Mathematical Society