# A syntactic approach to continuity of T-definable functionals

## TL;DR

This paper presents a new syntactic proof that all functions definable in G"odel's System T are continuous, using a translation into itself and formalization in Agda to compute moduli of continuity.

## Contribution

It introduces a novel syntactic translation and a continuity predicate, providing a formalized, executable proof of continuity for T-definable functions.

## Key findings

- Proof formalized in Agda
- Able to compute moduli of continuity
- New syntactic approach to continuity

## Abstract

We give a new proof of the well-known fact that all functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ which are definable in G\"odel's System T are continuous via a syntactic approach. Differing from the usual syntactic method, we firstly perform a translation of System T into itself in which natural numbers are translated to functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$. Then we inductively define a continuity predicate on the translated elements and show that the translation of any term in System T satisfies the continuity predicate. We obtain the desired result by relating terms and their translations via a parametrized logical relation. Our constructions and proofs have been formalized in the Agda proof assistant. Because Agda is also a programming language, we can execute our proof to compute moduli of continuity of T-definable functions.