Richardson, Dan Elimination of infinitesimal quantifiers. (English) Zbl 0972.03032 J. Pure Appl. Algebra 139, No. 1-3, 235-253 (1999). 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 Keywords:extension of the ordered field of the reals; totally defined analytic functions; locally Pfaffian; globally Noetherian; Whitney regularity; infinitesimal quantifiers PDFBibTeX XMLCite \textit{D. Richardson}, J. Pure Appl. Algebra 139, No. 1--3, 235--253 (1999; Zbl 0972.03032) Full Text: DOI References: [1] A. Gabrielov, Frontier and closure of a semi-Pfaffian set, Discrete Comput. Geom., in press.; A. Gabrielov, Frontier and closure of a semi-Pfaffian set, Discrete Comput. Geom., in press. · Zbl 0911.32009 [2] A. Gabrielov, Counterexamples to quantifier elimination for fewnomials and exponential expressions, Preprint.; A. Gabrielov, Counterexamples to quantifier elimination for fewnomials and exponential expressions, Preprint. · Zbl 1149.14044 [3] Gabrielov, A.; Vorobjov, N., Complexity of stratifications of semi-Pfaffian sets, Discrete Comput. Geom., 14, 71-91 (1995) · Zbl 0832.68056 [4] D. Richardson, ISSAC 1996, Solution of elementary systems of equations in a box in \(R^n\); D. Richardson, ISSAC 1996, Solution of elementary systems of equations in a box in \(R^n\) [5] Richardson, D., How to Recognize Zero, J. Symbolic Comput., 24, 627-645 (1997) · Zbl 0917.11062 [6] van den Dries, L.; Miller, C., Geometric categories and o-minimal structures, Duke Math. J., 84, 2, 497-540 (1996) · Zbl 0889.03025 [7] C.T.C. Wall, Regular Stratifications, Dynamical Systems, Lecture Notes in Mathematics, vol. 468, Springer, Berlin, 1974, pp. 332-344.; C.T.C. Wall, Regular Stratifications, Dynamical Systems, Lecture Notes in Mathematics, vol. 468, Springer, Berlin, 1974, pp. 332-344. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.