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A question of van den Dries and a theorem of Lipshitz and Robinson: not everything is standard. (English) Zbl 1118.03027
One of the most typical examples of o-minimal structures is the ordered field R of reals; but every real closed field, in particular every non-standard elementary extension of the reals, is o-minimal as well, and so are several notable expansions of R to larger languages. The paper under review deals with the relationship between these o-minimal expansions of the reals and the o-minimal expansions of arbitrary real closed fields. In particular it examines and answers negatively the following question of L. van den Dries: Let $$L$$ be a language extending the one $$\{ +, \, \cdot, \, < \}$$ of ordered rings by new relation and function symbols. Let $$\varphi$$ be a sentence in this language $$L$$. Suppose that $$\varphi$$ is true in every possible expansion of R in $$L$$. Is $$\varphi$$ necessarily true in every o-minimal $$L$$-expansion of a real closed field?
The counterexample is based on a recent construction of L. Lipshitz and Z. Robinson, producing a new o-minimal structure as the expansion of the real closed field $$R$$ of Puiseux series in $$t$$ over R, with $$t$$ an infinitesimal, by functions $$f_p(x_1, \, \ldots, \, x_n)$$ where $$p(x_1, \, \ldots, \, x_n)$$ ranges over formal power series over R: $$f_p$$ is the function corresponding to $$p$$ in the box $$[-t, \, t]^n$$ (where $$p$$ converges) and is defined as 0 outside this cube.
In this framework, the authors single out a property that can be written as a first-order sentence $$\varphi$$ in the language $$L$$ with two new 2-ary function symbols $$F_1$$, $$F_2$$, to be viewed as the real part and the imaginary part of a unique 1-ary function $$F$$ in the algebraic closure of the underlying real closed field. $$\varphi$$ establishes some differentiability condition on $$F$$. It is shown that $$\varphi$$ is false in every possible $$L$$-expansion of the reals, but is true in a certain $$L$$-expansion of $$R$$, built inside the Lipshitz-Saracino structure. In fact a suitable power series $$p$$ and the corresponding $$f_p$$ are involved to define the intepretation of $$F$$, and consequently of $$F_1$$ and $$F_2$$.

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 12L12 Model theory of fields
##### Keywords:
real closed field; o-minimal expansion
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##### References:
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