axiom of reducibility

The axiom of reducibility was an assumption introduced by Bertrand Russell and Alfred North Whitehead in their work Principia Mathematica. This axiom was necessary to bridge the gap between their "ramified" theory of types and the goal of deriving all of mathematics from logic.1

The axiom of reducibility implies a partial collapse of the ramified hierarchy, which was designed to avoid paradoxes by assigning logical types to entities. Henri Poincaré had objected to "impredicative constructions," which define an entity in terms of a collection that includes the entity itself. While Russell and Whitehead aimed for a predicative type theory, the axiom of reducibility ultimately rendered their system impredicative. Frank Ramsey later showed that the system could be revised without the axiom of reducibility by taking an extensional view of higher-order objects.12