Creative and Productive Sets - History

History

Emil Post's 1944 seminal paper defined the concept he called a Creative set. Reiterating, the 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. 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, essentialy creative."

The usual creative set defined using the diagonal function has its own historical development. Alan Turing in a 1936 article on the Turing machine showed the existence of a universal computer that computes the function. The function is defined such that (the result of applying the instructions coded by to the input ), and is universal in the sense that any calculable partial function is given by for all where codes the instructions for . Using the above notation, 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.