Recursive realizability

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A more precise definition of the intuitionistic semantics of arithmetical formulas, based on the concept of a partial recursive function as proposed by S.C. Kleene (see [1], [2]). For every closed arithmetical formula , a relation "the natural number e realizes the formula F" is defined; it is denoted by . The relation is defined inductively according to the structure of the formula .

1) If is an atomic formula without free variables, i.e. is of the form where and are constant terms, then if and only if and the values of the terms and coincide.

Let and be formulas without free variables.

2) if and only if , where , .

3) if and only if and , or and .

4) if and only if is the Gödel number of a unary partial recursive function such that for any natural number , implies that is defined at and .

5) if and only if .

Let be a formula without free variables other than ; if is a natural number, then is a term which denotes the number in formal arithmetic.

6) if and only if and .

7) if and only if is the Gödel number of a recursive function such that for any natural number the number realizes .

A closed formula is called realizable if there is a number which realizes . A formula containing the free variables can be regarded as a predicate in ( "the formula Ay1…ym is realizable" ). If a formula is derivable from realizable formulas in intuitionistic arithmetic, then is realizable (see [3]). In particular, every formula which can be proved in intuitionistic arithmetic is realizable. A formula can be given for which the formula is not realizable. Accordingly, in this case the formula is realizable although it is classically false.

Every predicate formula which is provable in intuitionistic predicate calculus has the property that each arithmetical formula obtained from by substitution is realizable. Predicate formulas possessing this property are called realizable. It has been shown [4] that the propositional formula:

where denotes the formula , is realizable but cannot be derived in intuitionistic propositional calculus.


[1] S.C. Kleene, "On the interpretation of intuitionistic number theory" J. Symbolic Logic , 10 (1945) pp. 109–124
[2] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
[3] D. Nelson, "Recursive functions and intuitionistic number theory" Trans. Amer. Math. Soc. , 61 (1947) pp. 307–368
[4] G.F. Rose, "Propositional calculus and realizability" Trans. Amer. Math. Soc. , 75 (1953) pp. 1–19
[5] P.S. Novikov, "Constructive mathematical logic from a classical point of view" , Moscow (1977) (In Russian)



[a1] M.J. Beeson, "Foundations of constructive mathematics" , Springer (1985)
[a2] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974)
How to Cite This Entry:
Recursive realizability. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article