Propositional Calculus and Realizability
نویسنده
چکیده
In [13], Kleene formulated a truth-notion called "readability" for formulas of intuitionistic number theory(1). David Nelson(2) showed that every number-theoretic formula deducible in the intuitionistic predicate calculus(3) (stated by means of schemata, without proposition or predicate variables) from realizable number-theoretic formulas is realizable. In particular, then, every formula in the symbolism of the predicate calculus (with proposition and predicate variables) which is provable in the intuitionistic predicate calculus has the property that every number-theoretic formula (in the number-theoretic symbolism without proposition or predicate variables) which comes from it by substitution is realizable. This property was applied by Kleene to demonstrate that certain formulas provable in the classical predicate calculus are unprovable in the intuitionistic predicate calculu's(4). The question naturally arises whether the converse of Nelson's result holds. That is to say, is an arbitrary formula of the predicate calculus provable whenever every number-theoretic formula obtained from it by substitution is realizable? If this question could be answered in the affirmative, we should have a completeness theorem for the intuitionistic predicate calculus(6).
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