Real closed exponential fields

Paola D'Aquino, et al. "Real closed exponential fields." Fundamenta Mathematicae 219.2 (2012): 163-190. <http://eudml.org/doc/286459>.

@article{PaolaDAquino2012,
abstract = {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 $L_\{ω₁^\{CK\}\}$.},
author = {Paola D'Aquino, Julia F. Knight, Salma Kuhlmann, Karen Lange},
journal = {Fundamenta Mathematicae},
keywords = {real closed exponential field; exponential integer part},
language = {eng},
number = {2},
pages = {163-190},
title = {Real closed exponential fields},
url = {http://eudml.org/doc/286459},
volume = {219},
year = {2012},
}

TY - JOUR
AU - Paola D'Aquino
AU - Julia F. Knight
AU - Salma Kuhlmann
AU - Karen Lange
TI - Real closed exponential fields
JO - Fundamenta Mathematicae
PY - 2012
VL - 219
IS - 2
SP - 163
EP - 190
AB - 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 $L_{ω₁^{CK}}$.
LA - eng
KW - real closed exponential field; exponential integer part
UR - http://eudml.org/doc/286459
ER -