On the decidability of the real exponential field. (English) Zbl 0896.03012

Macintyre, Angus; Wilkie, A. J.

Odifreddi, Piergiorgio (ed.), Kreiseliana: about and around Georg Kreisel. Wellesley, MA: A K Peters. 441-467 (1996).

The authors discuss the “two structures \(\langle\overline{\mathbb R}, \exp\rangle\) and \(\langle \overline {\mathbb R}, e\rangle\), where \(\overline {\mathbb R}= \langle {\mathbb R},+,\cdot, -, 0, 1, <\rangle\) denotes the ordered field of real numbers (in the language of ordered rings), exp : \( {\mathbb R} \rightarrow {\mathbb R}\) is the usual exponential function and \(e\) : \({\mathbb R}\rightarrow{\mathbb R}, x \mapsto \text{ exp}((1 + x^2)^{-1})\) is the restricted exponential function […]” (p. 441). The theories of these structures are denoted by \(T_{\text{exp}}\) and \(T_e\). The decidability of \(T_{\text{exp}}\) is finally proved. Good knowledge of Wilkie’s paper on model completeness of \(T_{\text{exp}}\) and \(T_e\) [A. J. Wilkie, J. Am. Math. Soc. 9, No. 4, 1051-1094 (1996; Zbl 0892.03013)] is required.
For the entire collection see [Zbl 0894.03002].

Reviewer: V.Peckhaus (Erlangen)

MSC:

03B25	Decidability of theories and sets of sentences
12L05	Decidability and field theory
03C62	Models of arithmetic and set theory
03C35	Categoricity and completeness of theories
12L12	Model theory of fields

Keywords:

decidability; real exponential field

Citations:

Zbl 0892.03013