We consider the Σ1-fragment of second-order logic over the vocabulary (+, ×, 0, 1, <, S1, ..., Sk), interpreted over the reals, where the predicate symbols Si are interpreted as semi- algebraic sets. We show that, in this context, satisfiability of formulas is decidable for the first-order ∃∗-quantifier fragment and undecidable for the ∃∗∀- and ∀∗-fragments. We also show that for these three fragments the same (un)decidability results hold for containment and equivalence of formulas.
