Joyal's arithmetic universes via type theory

André Joyal constructed arithmetic universes to provide a categorical proof of Gödel incompleteness results. He built them in three stages: he first took a Skolem theory, then the category of its predicates and finally he made the exact completion out of the latter. Here, we prove that the construction of an initial arithmetic universe is equivalent to that of an initial list-arithmetic pretopos and also of an initial arithmetic pretopos. The initial list-arithmetic pretopos is built out of its internal language formulated as a dependent typed calculus in the style of Martin-Löf’s extensional type theory. Analogously, we prove that the second stage of Joyal's construction is equivalent to taking an initial arithmetic lextensive category or an initial regular locos. We conclude by proposing the notion of list-arithmetic pretopos as the general definition of arithmetic universe. We are motivated from the fact in any list-arithmetic pretopos we can show the existence of free internal categories and diagrams as in any of Joyal's arithmetic universes.