  Poincaré, Russell and Weyl’s foundational analysis of predicativity has paved the way for a variety of logical analyses of the notion. The most commonly accepted analysis is due to Feferman and Schütte (independently) following lines indicated by Kreisel (Kreisel 1958, Feferman 1964 and Schütte 1965; 1965a). Here proof theory has played a pivotal role. In very rough terms, the idea was to single out a collection of theories (a transfinite progression of systems of ramified second order arithmetic indexed by ordinals) by means of which to characterise a certain notion of predicative ordinal. Feferman and Schütte’s proof theoretic analysis of these theories has identified an ordinal, usually referred to as \(\Gamma_0\), which is the least non-predicative ordinal according to this notion. A formal system is considered  predicatively justifiable if it is proof-theoretically reducible to a system of ramified second order arthmetic indexed by an ordinal less then \(\Gamma_0\). Therefore in proof theory \(\Gamma_0\) is usually considered as representing the limit of predicativity. (See Feferman 2005 for a more accurate informal account of this notion of predicativity and for further references. See also Crosilla 2017. The reader may also consult the section on  predicativism  in the entry on philosophy of mathematics and the entry on  paradoxes and contemporary logic).