Logicism III: Theory of Types

Can the theory of types serve as a logicist foundation of mathematics?

Hint: you might wish to discuss the significance of either the reducibility axiom, or the infinity axiom (or both).

Readings

Russell (1908);

(Weyl, 1946, 2-13, §§1-4);

Hatcher (1968);

(Feferman, 1964, 1-30, Part I)

Further readings

(Weyl, 1987, ch. I, §§4-8 and Conclusion);

(Gödel, 1944, 452-69);

(Kilmister, 1967, ch. 7)