Randomising Realizability

We consider a randomised version of Kleene’s realizability interpretation of intuitionistic arithmetic in which computability is replaced with randomised computability with positive probability. In particular, we show that (i) the set of randomly realizable statements is closed under intuitionistic first-order logic, but (ii) different from the set of realizable statements, that (iii) “realizability with probability 1” is the same as realizability and (iv) that the axioms of bounded Heyting’s arithmetic are randomly realizable, but some instances of the full induction scheme fail to be randomly realizable.