The logical consequence undisputedly is the central concept of logic. The main purpose of logic is to tell us what follows logically from what. Logical consequence is a relation between a given set of formulas and the formulas that logically follow. In this work, we inspect set theoritical properties of the logical consequence related to denumerable sets and independet sets formulas of classical logic.  We show that that intersections of independent sets are independent but union of two independent sets is not in general independent.