They are a standard topic in mathematical logic textbooks such as Soare (1987) and Rogers (1987).
is an admissible 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 total recursive (computable) function
A set A of natural numbers is called creative if A is recursively enumerable and its complement
Not every productive set has a recursively enumerable complement, however, as illustrated below.
, the set representing the halting problem.
To see this, we apply the definition of a productive function and show separately that
: No productive set A can be recursively enumerable, because whenever A contains every number in an r.e.
Similarly, no creative set can be decidable, because this would imply that its complement, a productive set, is recursively enumerable.
The following theorems, due to Myhill (1955), show that in a sense all creative sets are like
The following are equivalent: 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 seminal paper of Post (1944) defined the concept he called a Creative set.
referenced above and defined as the domain of the function
that takes the diagonal of all enumerated 1-place computable partial functions and adds 1 to them is an example of a creative set.
[2] Post gave a version of Gödel's Incompleteness Theorem using his creative sets, where originally Gödel had in some sense constructed a sentence that could be freely translated as saying "I am unprovable in this axiomatic theory."
However, Gödel's proof did not work from the concept of true sentences, and rather used the concept of a consistent theory, which led to the second incompleteness theorem.
After Post completed his version of incompleteness he then added the following: "The conclusion is unescapable that even for such a fixed, well defined body of mathematical propositions, mathematical thinking is, and must remain, essentially creative.
Alan Turing in a 1936 article on the Turing machine showed the existence of a universal computer that computes the
(the result of applying the instructions coded by
), and is universal in the sense that any calculable partial function
, and the diagonal function arises quite naturally as
Ultimately, these ideas are connected to Church's thesis that says the mathematical notion of computable partial functions is the correct formalization of an effectively calculable partial function, which can neither be proved or disproved.
Church used lambda calculus, Turing an idealized computer, and later Emil Post in his approach, all of which are equivalent.
Deborah Joseph and Paul Young (1985) formulated an analogous concept, polynomial creativity, in computational complexity theory, and used it to provide potential counterexamples to the Berman–Hartmanis conjecture on isomorphism of NP-complete sets.