The modified realizability topos

The Gleason Cover of a Realizability Topos

Recently Benno van den Berg [1] introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the ‘constant object’ functor from the topos of sets preserves finite coproducts, and that De Morgan’s law is satisfied. In this paper we show that these...

Bounded modified realizability

We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak Kön...

Relative and modified relative realizability

Modified Realizability and Inductive Types

In 1959, Kreisel introduced a notion of “modified” realizability that, among other things, provides an alternative technique to Gödel functional (dialectica) interpretation for establishing the connection between Peano Arithmetic and System T. While the dialectica interpretation has been widely studied in the literature, Kreisel’s technique, although remarkably simpler, has apparently been almo...

A partial analysis of modified realizability

In [3], the following semantical picture of modified realizability is sketched: Consider the topos Set, whose objects are functions between sets, and whose arrows are commutative squares. The logic of Set corresponds to “Kripke forcing over a 2-element linear order”. Over Set, one can define the standard realizability topos Eff→ (not to be confused with Eff!). The modified realizability topos, ...