Introduction to higher order categorical logic
Introduction to higher order categorical logic
Topology via logic
A Note on Powerdomains and Modalitiy
Proceedings of the 1983 International FCT-Conference on Fundamentals of Computation Theory
Reflection Into Models of Finite Decidable FP-sketches in an Arithmetic Universe
Electronic Notes in Theoretical Computer Science (ENTCS)
The dedekind reals in abstract stone duality
Mathematical Structures in Computer Science
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The first paper published on Abstract 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.