On the Computability of the Fan Functional

Abstract

We give a self-contained presentation of the original argument for why the fan functional is countably recursive, but not Kleene computable. This proof, from 1958, is due to the second author and has not been published before. A lemma concerning the Kleene computable functionals used in the proof is unnecessarily strong, which is fortunate since there is a counterexample, due to the first author, which we include.