  For constructive foundational theories a more “liberal” approach to predicativism has been suggested, starting from work in the late 1950’s of Lorenzen, Myhill and Wang (see e.g. Lorenzen and Myhill 1959). The driving idea is that so-called inductive definitions ought to be allowed in the realm of constructive mathematics. The intuitive justification of inductive definitions is related to the fact that they can be expressed by means of finite rules, in a “bottom-up” way. The proof-theoretic strength of theories of inductive definitions goes well beyond Feferman and Schütte’s bound (Buchholz, Feferman, Pohlers and Sieg 1981). Thus relatively strong theories are considered predicative in today’s foundations of constructive mathematics. This more liberal notion of predicativity has often been termed generalised predicativity. In this entry we  simply write predicativity for generalised predicativity and call  predicativity given the natural numbers the better known form of predicativity which arises in the classical context and was analysed by Kreisel, Feferman and Schütte.