### Recursive Predicates and Quantifiers

S. C. Kleene
1943 Transactions of the American Mathematical Society
This paper contains a general theorem on the quantification of recursive predicates, with applications to the foundations of mathematics. The theorem (Theorem II) is a slight extension of previous results on Herbrand-Gödel general recursive functions(2), while the applications include theorems of Church (Theorem VII)(3) and Gödel (Theorem VIII)(4) and other incompleteness theorems. It is thought that in this treatment the relationship of the results stands out more clearly than before. The
more » ... al theorem asserts that to each of an enumeration of predicate forms, there is a predicate not expressible in that form. The predicates considered belong to elementary number theory. The possibility that this theorem may apply appears whenever it is proposed to find a necessary and sufficient condition of a certain kind for some given property of natural numbers; in other words, to find a predicate of a given kind equivalent to a given predicate. If the specifications on the predicate which is being sought amount to its having one of the forms listed in the theorem, then for some selection of the given property a necessary and sufficient condition of the desired kind cannot exist. In particular, it is recognized that to find a complete algorithmic theory for a predicate P(a) amounts to expressing the predicate as a recursive predicate. By one of the cases of the theorem, this is impossible for a certain P(a), which gives us Church's theorem. Again, when we recognize that to give a complete formal deductive theory (symbolic logic) for a predicate P{a) amounts to finding an equivalent predicate of the form (Ex)R(a, x) where R(a, x) is recursive, we have immediately Gödel's theorem, as another case of the general theorem. Still another application is made, when we consider the nature of a constructive existence proof. It appears that there is a proposition provable classically for which no constructive proof is possible (Theorem X). The endeavor has been made to include a fairly complete exposition of definitions and results, including relevant portions of previous theory, so that