Abstract

The main result of this paper is the equivalence of several definition schemas of bar recursion occurring in the literature on functionals of finite type. We present the theory of functionals of finite type, in [T] denoted byqf-WE-HA ω, which is necessary for giving the equivalence proofs. Moreover we prove two results on this theory that cannot be found in the literature, namely the deduction theorem and a derivation of Spector's rule of extensionality from [S]: ifP→T 1=T 2 and Q[X∶≡T1], then P→Q[X∶≡ T2], from the at first sight weaker rule obtained by omitting “P→”