Realizability
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another.
Realizability can be seen as a formalization of the BHK interpretation of intuitionistic logic; in realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly provide.
Beyond giving insight into intuitionistic provability, realizability can be applied to prove the disjunction and existence properties for intuitionistic theories and to extract programs from proofs, as in proof mining. It is also related to topos theory via the realizability topos.
Kleene's original version of realizability uses natural numbers as realizers for formulas in Heyting arithmetic. The following clauses are used to define a relation "n realizes A" between natural numbers n and formulas A in the language of Heyting arithmetic. A few pieces of notation are required: first, an ordered pair (n,m) is treated as a single number using a fixed effective pairing function; second, for each natural number n, φn is the computable function with index n.
With this definition, the following theorem is obtained:
On the other hand, there are formulas that are realized but which are not provable in HA, a fact first established by Rose.
Further analysis of the method can be used to prove that HA has the "disjunction and existence properties":
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