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The aim of this paper is to prove that every subset of R definable from addition, multiplication and exponentiation admits a stratification satisfying Whitney’s conditions a) and b). 1. Preliminaries. Let An be the smallest ring of real-valued functions on R such that: (a) An contains all polynomials, i.e. R[x1, . . . , xn] ⊂ An. (b) An is closed under taking exponentiation, i.e. if f ∈ An, then exp f ∈ An. 1.1. Definition. Let D = (Dn)n∈N be the smallest class of subsets of Euclidean spaces R, n ∈ N, where Dn is a class of subsets of R, satisfying the following properties for all n: (D1) Dn contains all sets of the form {x ∈ R : f(x) = 0}, where f ∈ An. (D2) If S, T ∈ Dn, then S ∪ T , S ∩ T and S \ T ∈ Dn. (D3) If S ∈ Dn+1, then π(S) ∈ Dn, where π : R → R is the natural projection. A set S is called a Dn-set iff S ∈ Dn. A D-set is a Dn-set for some n ∈ N. A function f : S → R is called a D-function iff its graph is a D-set. R e m a r k. The class D contains all semi-algebraic sets. A D-set, in general, is not subanalytic (e.g. {(x, y) : x > 0, y = exp(−1/x)}). If f is a D-function, then so is exp f . If, in addition, f > 0, then log f , f (α ∈ R) are D-functions. The closure, the interior and the boundary in R of a Dn-set are Dn-sets. The following theorem is due to Wilkie [9], [10], which is an essential result for the class D. 1991 Mathematics Subject Classification: 14P99, 32B25.

Whitney stratification of sets definable in the structure $ℝ_{exp}$