Representing definable functions of $\mathrm{HA}^ω$ 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 ($\mathrm{HA}^ω$) 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 $\mathrm{HA}^ω$ 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 $\mathrm{HA}^ω$ with the required property. As applications of this result, we prove some well-known properties of $\mathrm{HA}^ω$: uniform continuity of definable functions from $\mathbb{N}^{\mathbb{N}}$ to $\mathbb{N}$ 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.