The article investigates a system of polymorphically typed combinatory logic which is equivalent to Gödel’s T. A notion of (strong) reduction is defined over terms of this system and it is proved that the class of well-formed terms is closed under both bracket abstraction and reduction. The main new result is that the number of contractions needed to reduce a term to normal form is computed by an ε 0-recursive function. The ordinal assignments used to obtain this result are also (...) used to prove that the system under consideration is indeed equivalent to Gödel’s T. It is hoped that the methods used here can be extended so as to obtain similar results for stronger systems of polymorphically typed combinatory terms. An interesting corollary of such results is that they yield ordinally informative proofs of normalizability for sub-systems of second-order intuitionist logic, in natural deduction style. (shrink)

This article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy (...) of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions. (shrink)