 Creative and productive sets

In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic. They are a standard topic in mathematical logic textbooks such as Soare (1987) and Rogers (1987).
Contents
Definition and example
For the remainder of this article, assume that φ_{i} is an acceptable numbering of the computable functions and W_{i} the corresponding numbering of the recursively enumerable sets.
A set A of natural numbers is called productive if there exists a partial computable function f so that for all , if then The function f is called the productive function for A.
A set A of natural numbers is called creative if A is recursively enumerable and its complement is productive. Not every productive set has a recursively enumerable complement, however, as illustrated below.
The archetypal creative set is , the set representing the halting problem. Its complement is productive with productive function f(i) = i. To see this, assume . If i were in W_{i} then and thus . This would mean , so we can conclude , which means .
Properties
No productive set A can be recursively enumerable, because whenever A contains every number in an r.e. set W_{i} it contains other numbers, and moreover there is an effective procedure to produce an example of such a number from the index i. Similarly, no creative set can be decidable, because this would imply that its complement, a productive set, is recursively enumerable.
Any productive set has a productive function that is injective and total.
The following theorems, due to Myhill (1955), show that in a sense all creative sets are like K and all productive sets are like (see Soare (1987) and Rogers (1987)).
Theorem. Let P be a set of natural numbers. The following are equivalent:
 P is productive.
 is 1reducible to P.
 is mreducible to P.
Theorem. Let C be a set of natural numbers. The following are equivalent:
 C is creative.
 C is 1complete
 C is recursively isomorphic to K, that is, there is a total computable bijection f on the natural numbers such that f(C) = K.
Applications in mathematical logic
The set of all provable sentences in an effective axiomatic system is always a recursively enumerable set. If the system is suitably complex, like firstorder arithmetic, then the set T of Gödel numbers of true sentences in the system will be a productive set, which means that whenever W is a recursively enumerable set of true sentences, there is at least one true sentence that is not in W. This can be used to give a rigorous proof of Gödel's first incompleteness theorem, because no recursively enumerable set is productive. The complement of the set T will not be recursively enumerable, and thus T is an example of a productive set whose complement is not creative.
References
 Davis, M., 1982. Computability and Unsolvability. New York: Dover.
 Kleene, S. C.,2002. Mathematical Logic. New York: Dover.
 Myhill, J., 1955. "Creative sets," Z. Math. Logik Grundlag. Math., v. 1, pp. 97–108.
 Rogers, H., 1987. Theory of Recursive Functions and Effective Computability. Cambridge, MA: MIT Press.
 Soare, R., 1987. Recursively Enumerable Sets and Degrees. Berlin: Springer.
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