Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe

The first paper published on Stone Duality showed that the overt discrete objects (those admitting ∃ and = internally) form a pretopos, i.e. a category with finite limits, stable disjoint coproducts and stable effective quotients of equivalence relations. Using an N-indexed least fixed point axiom, here we show that this full subcategory is an arithmetic universe, having a free semilattice (“collection of Kuratowski-finite subsets”) and a free monoid (“collection of lists”) on any overt discrete object. Each finite subset is represented by its pair (□, ◊) of modal operators, although a tight correspondence with these depends on a stronger Scott-continuity axiom. Topologically, such subsets are both compact and open and also involve proper open maps. In applications of ASD this can eliminate lists in favour of a continuation-passing interpretation.