On Krivine's Realizability Interpretation of Classical Second-Order Arithmetic

This article investigates Krivine’s realizability interpretation of classical second-order arithmetic and its recent extension handling countable choice. We will start by presenting a two-step interpretation which first eliminates classical logic via a negative translation and then applies standard realizability interpretation. We then argue that a slight variant of Krivine’s interpretation corresponds to this two-step interpretation. This variant can be viewed as the continuation passing style variant of Krivine’s original interpretation, and as such uses standard λ-terms and avoids the use of new continuation constants in the interpretation of classical logic.