## Real Closed Exponential Fields

2011
D'Aquino, Paola
Knight, Julia F.
Lange, Karen
Preprint
##### Abstract
In an extended abstract Ressayre considered real closed exponential fields and integer parts that respect the exponential function. He outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre's construction, which becomes canonical once we fix the real closed exponential field, a residue field section, and a well ordering of the field. The procedure is constructible over these objects; each step looks effective, but may require many steps. We produce an example of an exponential field $R$ with a residue field $k$ and a well ordering $<$ such that $D^c(R)$ is low and $k$ and $<$ are $\Delta^0_3$, and Ressayre's construction cannot be completed in $L_{\omega_1^{CK}}$.
510 Mathematics
##### Cite This
ISO 690D'AQUINO, Paola, Julia F. KNIGHT, Salma KUHLMANN, Karen LANGE, 2011. Real Closed Exponential Fields
BibTex
@unpublished{DAquino2011Close-21245,
year={2011},
title={Real Closed Exponential Fields},
author={D'Aquino, Paola and Knight, Julia F. and Kuhlmann, Salma and Lange, Karen},
note={Also publ. in: Fundamenta Mathematicae ; 219 (2012), 2. - S. 163-190}
}

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<dcterms:abstract xml:lang="eng">In an extended abstract Ressayre considered real closed exponential fields and integer parts that respect the exponential function. He outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre's construction, which becomes canonical once we fix the real closed exponential field, a residue field section, and a well ordering of the field. The procedure is constructible over these objects; each step looks effective, but may require many steps. We produce an example of an exponential field $R$ with a residue field $k$ and a well ordering $&lt;$ such that $D^c(R)$ is low and $k$ and $&lt;$ are $\Delta^0_3$, and Ressayre's construction cannot be completed in $L_{\omega_1^{CK}}$.</dcterms:abstract>
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##### Comment on publication
Also publ. in: Fundamenta Mathematicae ; 219 (2012), 2. - S. 163-190
Yes