The Herbrand Topos

We define a new topos, the Herbrand topos, inspired by the modified realizability topos and our earlier work on Herbrand realizability. We also introduce the category of Herbrand assemblies and characterise these as the double-negation-separated objects in the Herbrand topos. In addition, we show that the category of sets is included as the category of double-negation-sheaves and prove that the inclusion functor preserves and reflects validity of first-order formulas.