Elimination of infinitesimal quantifiers Original Research Article

Infinitesimal quantifiers are of the form (there existsXnot, vert, similar0)A, meaning (for allη>0)(there existsX)(X<ηlogical andA); and (for allXnot, vert, similar0)A, meaning (there existsη>0)(for allX)(X<η→A). 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.