Realizability

Realizability is a part of proof theory which can be used to handle information about formulas instead of about the proofs of formulas. A natural number n is said to realize a statement in the language of arithmetic of natural numbers. Other logical and mathematical statements are also realizable, providing a method for interpreting well formed formulas without resorting to proofs for arriving at those formulas.

Origins
Kleene introduced the concept of realizability in 1945 in the hopes of it being a faithful mirror of intuitionistic reasoning, but this conjecture was first disproved by Rose with his example of realizable propositional  formulas that are unprovable in intuitionist calculus. Realizability appears to defy axiomatization due to its complexity, but it may be approachable through a higher-order Heyting arithmetic (HA). For HA3, a completeness property for the  category of modest sets may be proved from the  axioms which characterize the realizability of HA3.

Later developments
Relative realizability is an intuitionist analysis of recursive or recursively enumerable elements of data structures that are not necessarily computable, such as computable operations on all real numbers when reals can be only approximated on digital computer systems.