Publikation: Real closed exponential fields
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2012
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Fundamenta Mathematicae. 2012, 219(2), pp. 163-190. ISSN 0016-2736. eISSN 1730-6329. Available under: doi: 10.4064/fm219-2-6
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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 Dc(R) is low and k and ≺ are Δ 0 3, and Ressayre's construction cannot be completed in LωCK1.
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D'AQUINO, Paola, Julia F. KNIGHT, Salma KUHLMANN, Karen LANGE, 2012. Real closed exponential fields. In: Fundamenta Mathematicae. 2012, 219(2), pp. 163-190. ISSN 0016-2736. eISSN 1730-6329. Available under: doi: 10.4064/fm219-2-6BibTex
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year={2012},
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title={Real closed exponential fields},
number={2},
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journal={Fundamenta Mathematicae},
pages={163--190},
author={D'Aquino, Paola and Knight, Julia F. and Kuhlmann, Salma and Lange, Karen}
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<dcterms:abstract xml:lang="eng">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<sup>c</sup>(R) is low and k and ≺ are Δ 0 3, and Ressayre's construction cannot be completed in L<sub>ωCK1</sub>.</dcterms:abstract>
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