Real numbers in the topos of sheaves over the category of filters

AbstractThe sheaves over the category of filters, with the precanonical topology, serve as a universe of sets where nonstandard analysis can be developed along constructive principles. In this paper we show that the Dedekind real numbers of this topos can be characterised as the nonstandard hull of the rational numbers. Moreover, it is proved that the axiom of choice holds on standard sets of the topos