# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Real closed exponential fields

### Tom 219 / 2012

Fundamenta Mathematicae 219 (2012), 163-190 MSC: Primary 03C57; Secondary 03C60, 03C70. DOI: 10.4064/fm219-2-6

#### Streszczenie

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

#### Autorzy

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

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