Real closed exponential fields. (English) Zbl 1285.03036

D’Aquino, Paola; Knight, Julia F.; Kuhlmann, Salma; Lange, Karen

Fundam. Math. 219, No. 2, 163-190 (2012).

Summary: J. P. Ressayre considered real closed exponential fields and “exponential” integer parts, i.e., integer parts that respect the exponential function. In [Oxf. Logic Guides 23, 278–288 (1993; Zbl 0791.03018)], 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 \(L_{\omega_1^{\mathrm{CK}}}\).

MSC:

03C60	Model-theoretic algebra
12L12	Model theory of fields

Keywords:

real closed exponential field; exponential integer part

Citations:

Zbl 0791.03018