A question of van den Dries and a theorem of Lipshitz and Robinson: not everything is standard.

*(English)*Zbl 1118.03027One 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 \(F_2\).

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 \(F_2\).

Reviewer: Carlo Toffalori (Camerino)

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\textit{E. Hrushovski} and \textit{Y. Peterzil}, J. Symb. Log. 72, No. 1, 119--122 (2007; Zbl 1118.03027)

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