Duality, non-standard elements, and dynamic properties of r.e. sets

We investigate the Priestley dual (E⁎)⋆(E⁎)⋆ of the lattice E⁎E⁎ of r.e. sets modulo finite sets. Connections with non-standard elements of r.e. sets in models of 1st order true arithmetic as well as with dynamic properties of r.e. sets are pointed out. Illustrations include the Harrington–Soare dynamic characterization of small subsets, a model-theoretic characterization of promptly simple sets, and relations between the inclusion ordering of prime filters on E⁎E⁎ (a.k.a. points of (E⁎)⋆(E⁎)⋆) and the complexity of their index sets.