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.
Read more about this topic: Creative And Productive Sets
Famous quotes containing the word history:
“Bias, point of view, fury—are they ... so dangerous and must they be ironed out of history, the hills flattened and the contours leveled? The professors talk ... about passion and point of view in history as a Calvinist talks about sin in the bedroom.”
—Catherine Drinker Bowen (1897–1973)
“The history of the world is the record of the weakness, frailty and death of public opinion.”
—Samuel Butler (1835–1902)
“Let us not underrate the value of a fact; it will one day flower in a truth. It is astonishing how few facts of importance are added in a century to the natural history of any animal. The natural history of man himself is still being gradually written.”
—Henry David Thoreau (1817–1862)