Real closed exponential fields

Paola D'Aquino¹ ; Julia F. Knight² ; Salma Kuhlmann³ ; Karen Lange⁴

¹ Dipartimento di Matematica Seconda Università degli Studi di Napoli Viale Lincoln, 5 81100 Caserta, Italia
² Department of Mathematics University of Notre Dame 255 Hurley Hall Notre Dame, IN 46556, U.S.A.
³ Fachbereich Mathematik und Statistik Universität Konstanz Universitätsstraße 10 78457 Konstanz, Germany
⁴ Department of Mathematics Wellesley College 106 Central Street Wellesley, MA 02481, U.S.A.

Fundamenta Mathematicae, Tome 219 (2012) no. 2, pp. 163-190.

Résumé

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 $L_{\omega _1^{\rm CK}}$.

Détail

DOI : 10.4064/fm219-2-6

Keywords: ressayre considered real closed exponential fields exponential integer parts integer parts respect exponential function outlined proof every real closed exponential field has exponential integer part present paper detailed account ressayres construction analyze complexity ressayres construction canonical once fix real closed exponential field residue field section nbsp ordering prec construction clearly constructible these objects each step looks effective there may many steps produce example exponential field residue field section ordering prec low and prec delta ressayres construction cannot completed omega