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## Fundamenta Mathematicae

2012 | 219 | 2 | 163-190
Tytuł artykułu

### Real closed exponential fields

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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 ≺ 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 ≺ on R such that $D^{c}(R)$ is low and k and ≺ are Δ⁰₃, and Ressayre's construction cannot be completed in $L_{ω₁^{CK}}$.
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163-190
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wydano
2012
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autor
• Dipartimento di Matematica, Seconda Università degli Studi di Napoli, Viale Lincoln, 5, 81100 Caserta, Italia
autor
• Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556, U.S.A.
autor
• Fachbereich Mathematik und Statistik, Universität Konstanz, Universitätsstraße 10, 78457 Konstanz, Germany
autor
• Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA 02481, U.S.A.
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