We present three ordinal notation systems representing ordinals below ε 0 in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda.
We give a new proof of the well-known fact that all functions (N → N) → 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 (N → N) → 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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