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A generalization of the Tarski-Seidenberg theorem, and some nondefinability results. (English) Zbl 0612.03008
For the reals R, a semialgebraic subset of \(R^ m\) is a finite union of sets of the form \(\{x\in R^ m:\) \(p(x)=0\), \(q_ 1(x)>0,...,q_ k(x)>0\}\) where \(p,q_ 1,...,q_ k\in R[x]\). One consequence of Tarski’s decision procedure for the real field is that the definable subsets of \(<R,+,\cdot >\) are precisely the semialgebraic sets.
From the author’s introduction: ”Below we extend the system of semialgebraic sets in such a way that the Tarski-Seidenberg property, i.e., closure under definability and the topological finiteness phenomena are preserved. The polynomial growth property of semialgebraic functions is also preserved. This extended system contains the arctangent function on R, the sine function on any bounded interval, the exponential function \(e^ x\) on any bounded interval, but not the exponential function on all of R.... As a corollary we obtain that neither the exponential function on R, nor the set of integers is definable from addition, multiplication, and the restrictions of the sine and exponential functions to bounded intervals.”
The above mentioned extension is to the class of finitely subanalytic sets in \(R^ m\) (m\(\geq 1)\). The actual definition of this class requires several steps. But interestingly sets which are finite unions of sets of the form \(\{\) \(y\in U:\) \(f(y)=0\), \(g_ 1(y)>0,...,g_ k(y)>0\}\), where \(U\subseteq R^ m\) is open and \(f,g_ 1,...,g_ k\) are (real) analytic on U, play a basic role.
0-minimal Tarski systems on R are defined. The piecewise linear sets, the semialgebraic sets, and the finitely subanalytic sets are each examples of such a system. Certain finiteness results hold in all 0-mininal systems.
Reviewer: J.M.Plotkin

MSC:
03B25 Decidability of theories and sets of sentences
03C65 Models of other mathematical theories
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