Definability and Undefinability with Real Order at the Background

Let R be the integer order, that is the set of real numbers together with the standard order of reals. Let I be the set of integer numbers, let Y range over subsets of I, let P(I,X) be a monadic second-order formula about R, and let F be the collection of all subsets X of I such that P(I,X) holds in R. Even though F is a collection of subsets of I, its definition may involve quantification over reals and over sets of reals. In that sense, F is defined with the background of real order. Is that background essential or not? Maybe there is a monadic second-order formula Q(X) about I that defines F (so that F is the collection of all subsets X of I such that Q(X) holds in I). We prove that this is indeed the case, for any monadic second-order formula P(I,X). The claim remains true if the set I of integers is replaced above with any closed subset of R. The claim fails for some open subsets.