Summary: We compute the levels of complexity in analytical and arithmetical hierarchies for the sets of the $Σ$-formulas defining in the hereditarily finite superstructure over the ordered field of the reals the classes of open, closed, clopen, nowhere dense, dense subsets of $\Bbb R^n$, first category subsets in $\Bbb R^n$ as well as the sets of pairs of $Σ$-formulas corresponding to the relations of set equality and inclusion which are defined by them. It is also shown that the complexity of the set of the $Σ$-formulas defining connected sets is at least $Π_1^1$.