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Elimination of infinitesimal quantifiers. (English) Zbl 0972.03032

Summary: Infinitesimal quantifiers are of the form \((\exists x\sim 0)\varphi\), meaning \((\forall\varepsilon> 0)(\exists x)(|x|< \varepsilon\wedge\phi)\); and \((\forall x\sim 0)\varphi\), meaning \(\neg(\exists x\sim 0)\neg\varphi\). In the case of an extension of the ordered field of the reals obtained by adding totally defined analytic functions which are both locally Pfaffian and globally Noetherian (such as sine, cosine, exp), these quantifiers can be eliminated, as a consequence of work of A. Gabrielov on the closure of semi-Pfaffian sets. It is shown that Whitney regularity can be expressed using infinitesimal quantifiers. This can be used to obtain Whitney stratifications for semi-Pfaffian and sub-Pfaffian sets in this case.

MSC:

03C10 Quantifier elimination, model completeness, and related topics
14P15 Real-analytic and semi-analytic sets
03C80 Logic with extra quantifiers and operators
03C60 Model-theoretic algebra
14P25 Topology of real algebraic varieties
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