## Journal of Symbolic Logic

### On O-Minimal Expansions of Archimedean Ordered Groups

#### Abstract

We study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers $\mathscr{R}$. We then show that a definable function in an o-minimal expansion of $\mathscr{R}$ enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear expansion of $\mathscr{R}$. Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group.

#### Article information

Source
J. Symbolic Logic, Volume 60, Issue 3 (1995), 817-831.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744807

Mathematical Reviews number (MathSciNet)
MR1348995

Zentralblatt MATH identifier
0841.03020

JSTOR