Surreal fields stable under exponential and logarithmic functions

Abstract

Surreal numbers, have a very rich and elegant theory. This class of numbers, denoted by No, includes simultaneously the ordinal numbers and the real numbers, and forms a universal huge real closed field: It is universal in the sense that any real closed field can be embedded in it. Following Gonshor, surreal numbers can also be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. No can actually also be seen as an elegant particular (generalized) power series field with real coefficients, namely Hahn series with exponents in No itself. It can also be considered as a particular field of transseries, providing tools to do some analysis and asymptotic analysis for functions over the continuum, providing natural concepts for discussing hyperexponential or sublogarithm functions, and their asymptotics. In this article, we consider stability of subfields of No under exponential and logarithmic functions. Namely, we consider the set surreal numbers whose signs sequences have length less than some ordinal {\lambda}. Extending the discussion from van den Dries and Ehrlich, we show that is stable by exponential and logarithm iff {\lambda} is some {\epsilon}-number. Motivated in a longer term by computability issues using ordinal machines, we consider subfields stables by exponential and logarithmic functions defined by Hahn series that does not require to go up to cardinal lengths and exponents. We prove that No can be expressed as a strict hierarchy of subfields stable by exponential and logarithmic functions. This provides many explicit examples of subfields of No stable by exponential and logarithmic functions, and does not require to go up to a cardinal {\lambda} to provide such examples.