Given that the system of PM contains a ramified theory of types, however, the move to discussion of classes for the remainder of the work after ∗20 requires a further axiom, the axiom of reducibility, in order to allow a simple theory of types of classes. Consider the fundamental notion from the theory of real numbers of the least upper bound (l.u.b.) of a bounded class of real numbers. Consider the class of all real numbers whose square is less than or equal to 2, i.e., \(\{ x \mid x^2 \leq 2\}