  The axiom of reducibility draws attention to the problematic status of impredicative definitions. To quote Weyl 1946, the axiom of reducibility “is a bold, an almost fantastic axiom; there is little justification for it in the real world in which we live, and none at all in the evidence on which our mind bases its constructions”! So far, no contradictions have been found using the reducibility axiom. However, as we shall see below, proof-theoretic investigations confirm the extreme strength of such a principle.