Higher-Order Functions and Brouwer's Thesis

11 Aug 2016 Jonathan Sterling

Extending Mart\'in Escard\'o's effectful forcing technique, we give a new proof of a well-known result: Brouwer's monotone bar theorem holds for any bar that can be realized by a functional of type $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ in G\"odel's System T. Effectful forcing is an elementary alternative to standard sheaf-theoretic forcing arguments, using ideas from programming languages, including computational effects, monads, the algebra interpretation of call-by-name ${\lambda}$-calculus, and logical relations. Our argument proceeds by interpreting System T programs as well-founded dialogue trees whose nodes branch on a query to an oracle of type $\mathbb{N}\to\mathbb{N}$, lifted to higher type along a call-by-name translation... (read more)

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