Predicativity

Predicativity is a constraint on mathematical definition requiring that an object cannot be defined by reference to a totality of which it is already a member. A definition is impredicative if it defines an object by quantifying over a collection that includes the object being defined — a circularity that Henri Poincaré and Bertrand Russell identified as the source of the paradoxes (including Russell's paradox) that infected naive set theory.

The predicativity constraint was codified most precisely by Hermann Weyl in Das Kontinuum (1918) and subsequently by Solomon Feferman and Kurt Schütte, who independently identified the same ordinal — now called the Feferman-Schütte ordinal Γ₀ — as the precise boundary of predicative mathematics. Any proof-theoretic system with ordinal below Γ₀ reasons predicatively; systems that exceed Γ₀ commit to impredicative principles. Most of classical analysis, including theorems about completeness and fixed points, requires impredicativity: these theorems cannot be proved without defining objects by reference to totalities they belong to.

The philosophical weight of predicativity is considerable. It marks the boundary between constructive, step-by-step mathematical reasoning and the more powerful but philosophically contested methods of classical mathematics. That Γ₀ can be precisely identified means the boundary is not vague — it is a hard line in the proof-theoretic ordinal hierarchy. See Ordinal Analysis.