True V or not True V, That is the Question

In this paper we intend to argue that: (1) the question `True V or not True V' is central to both the philosophical and mathematical investigations of the foundations of mathematics; (2) when posed within a framework in which set theory is seen as a science of objects, the question `True V or not True V' generates a dilemma each horn of which turns out to be unacceptable; (3) a plausible way out of the dilemma mentioned at (2) is provided by an approach to set theory according to which this is considered to be a science of structures.