Informal Classical and Intuitionistic Proofs Together

I propose a fictional-factual interpretation of first-order arithmetic compatible both with classical logic and with the intuitionistic philosophical perspective. That should make the classical truth value of any arithmetical sentence accessible to the intuitionist. Using such a device, I explore the interplay between informal classical and intuitionistic proofs. I argue that the Gödel incompleteness theorems suggest a similarity between classical and intuitionistic informal proofs much stricter than it may appear at first sight. As a consequence, I try to show how intuitionistic reasoning can endorse some classical intuitive proofs.