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arXiv · 1904.09794

A syntactic approach to continuity of T-definable functionals

Chuangjie Xu
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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 Gödel'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.

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Chuangjie Xu. 2020-02-19. A syntactic approach to continuity of T-definable functionals. https://doi.org/10.23638/lmcs-16(1%3A22)2020

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