Real closed exponential fields

Tom 219 / 2012

Paola D'Aquino, Julia F. Knight, Salma Kuhlmann, Karen Lange Fundamenta Mathematicae 219 (2012), 163-190 MSC: Primary 03C57; Secondary 03C60, 03C70. DOI: 10.4064/fm219-2-6


Ressayre considered real closed exponential fields and “exponential” integer parts, i.e., integer parts that respect the exponential function. In 1993, 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 and then analyze the complexity. Ressayre's construction is canonical once we fix the real closed exponential field $R$, a residue field section $k$, and a well ordering $\prec $ on $R$. The construction is clearly constructible over these objects. Each step looks effective, but there may be many steps. We produce an example of an exponential field $R$ with a residue field section $k$ and a well ordering $\prec $ on $R$ such that $D^c(R)$ is low and $k$ and $\prec $ are $\Delta ^0_3$, and Ressayre's construction cannot be completed in $L_{\omega _1^{\rm CK}}$.


  • Paola D'AquinoDipartimento di Matematica
    Seconda Università degli Studi di Napoli
    Viale Lincoln, 5
    81100 Caserta, Italia
  • Julia F. KnightDepartment of Mathematics
    University of Notre Dame
    255 Hurley Hall
    Notre Dame, IN 46556, U.S.A.
  • Salma KuhlmannFachbereich Mathematik und Statistik
    Universität Konstanz
    Universitätsstraße 10
    78457 Konstanz, Germany
  • Karen LangeDepartment of Mathematics
    Wellesley College
    106 Central Street
    Wellesley, MA 02481, U.S.A.

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