Representing definable functions of HAω by neighbourhood functions

Abstract

Brouwer (1927) claimed that every function from the Baire space to natural numbers is induced by a neighbourhood function whose domain admits bar induction. We show that Brouwer's claim is provable in Heyting arithmetic in all finite types (HAω) for definable functions of the system. The proof does not rely on elaborate proof theoretic methods such as normalisation or ordinal analysis. Instead, we internalise in HAω the dialogue tree interpretation of Gödel's system T due to Escardó (2013). The interpretation determines a syntactic translation of terms, which yields a neighbourhood function from a closed term of HAω with the required property. As applications of this result, we prove some well-known properties of HAω: uniform continuity of definable functions from NN to N on the Cantor space; closure under the rule of bar induction; and closure of bar recursion for the lowest type with a definable stopping function.