Creative and productive sets

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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).

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[edit] Definition and example

For the remainder of this article, assume that \varphi_i is an acceptable numbering of the computable functions and Wi 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 i \in \mathbb{N}, if W_i \subseteq A then f(i) \in A \setminus W_i. 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 \mathbb{N}\setminus A is productive. Not every productive set has a recursively enumerable complement, however, as illustrated below.

The archetypal creative set is K = \{ i \mid i \in W_i \}, the set representing the halting problem. Its complement \bar{K} = \{ i \mid i \not \in W_i \} is productive with productive function f(i) = i. To see this, assume W_i \subseteq \bar{K}. If i were in Wi then  i \in K and thus  i \not \in \bar{K}. This would mean W_i \not \subseteq \bar{K}, so we can conclude i \not \in W_i, which means i \in \bar{K}.

[edit] Properties

No productive set A can be recursively enumerable, because whenever A contains every number in an r.e. set Wi 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 \bar{K} (see Soare (1987) and Rogers (1987)).

Theorem. Let P be a set of natural numbers. The following are equivalent:

Theorem. Let C be a set of natural numbers. The following are equivalent:

[edit] 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 first-order 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.

[edit] 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.