The Field of Reals with Multisummable Series and the Exponential Function

We begin by rephrasing a few de®nitions concerning model completeness and o-minimality in a setting suitable for this paper. This allows us to state our ®rst main result, Theorem A, in a way that is at the same time precise and widely accessible. Let F be a collection of functions f : R ! R for various m 2N :ˆ f0; 1; 2; . . .g. A set A Í R is said to be 0-de®nable in the structure RF :ˆ …R; < ; 0; 1; ‡; ÿ; ? ; F† (the expansion of the ordered ®eld of real numbers by F) if S belongs to the smallest collection of subsets of the spaces R :ˆ f0g, R ˆ R; R; R; . . . which