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SURREAL ORDERED EXPONENTIAL FIELDS

Part of: Ordered structures Model theory Ordered sets Topological fields

Published online by Cambridge University Press:  13 August 2021

PHILIP EHRLICH  and
ELLIOT KAPLAN
PHILIP EHRLICH
Affiliation:
DEPARTMENT OF PHILOSOPHY OHIO UNIVERSITYATHENS, OH 45701, USAE-mail:ehrlich@ohio.edu
ELLIOT KAPLAN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNURBANA, IL 61801, USAE-mail:eakapla2@illinois.edu
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Abstract

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$ -vector space) to be isomorphic to an initial subfield ( $K$ -subspace) of ${\mathbf {No}}$ , i.e. a subfield ( $K$ -subspace) of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$ . In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $({\mathbf {No}}, \exp )$ . These include all models of $T({\mathbb R}_W, e^x)$ , where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$ , which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R}((\omega ))^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $ .

Keywords

surreal numbersordered exponential fieldsreal closed exponential fieldstransseries

MSC classification

Primary: 06A05: Total order 03C64: Model theory of ordered structures; o-minimality
Secondary: 12J15: Ordered fields 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces 06F25: Ordered rings, algebras, modules
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 3 , September 2021 , pp. 1066 - 1115
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Copyright
© Association for Symbolic Logic 2021

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Footnotes

*

The in-text reference citations have been corrected. An erratum detailing this update has also been published (doi: 10.1017/jsl.2022.5).

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