Problem
Research questions and friction points this paper is trying to address.
Gödel's System T
monadic translation
logical relations
nucleus
continuity moduli
The Gentzen-style monadic translation of G\"odel's System T revisited
🤖 AI Summary
This study addresses the absence of a unified correctness proof and the limited extensibility of Gentzen-style monadic translations for Gödel's System T by reconstructing the translation via parameterized nuclei. Methodologically, it introduces a nuclear structure that does not require satisfaction of the monad laws, thereby simplifying the construction of internal dialogue trees. By integrating logical relations, monad theory, Church encodings, and higher-order recursion, the work establishes a unified correctness argument through the fundamental theorem of logical relations, subsequently extending the translation to sum types and finite lists. The primary contributions include a systematic comparison with Kuroda-style translations and a uniform representation within T of key concepts such as upper bounds and moduli of continuity, significantly enhancing the consistency and conciseness of the theoretical framework.
📝 Abstract
We revisit the Gentzen-style monadic translation of G\"odel's System T. The translation is parametrized by a nucleus, a monad-like structure that need not satisfy the monad laws. A fundamental theorem of logical relations provides a uniform correctness argument for its instances. By choosing suitable nuclei, we obtain majorants, moduli of pointwise and uniform continuity, internal dialogue trees, and functionals of general bar recursion, all represented by terms of T. The internal dialogue-tree application gives a simpler construction, with correctness established directly using Church-encoded terms. We also extend the translation and its fundamental theorem to sums and finite lists. Finally, we develop the Kuroda-style translation and its continuity instance, comparing the moduli obtained with those of the Gentzen-style translation.
Innovation
Methods, ideas, or system contributions that make the work stand out.
monadic translation
Gödel's System T
nucleus
logical relations
bar recursion
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