Computable Numbers

 was the title of a scientific paper by Alan Turing, where he disproved David Hilbert's thesis that all mathematical problems were solvable.

Turing worked on this paper for over a year and published it before World War II. The proof used the concept of a Universal Machine. (PROSE: The Turing Test)

Behind the scenes
Hilbert's program was an attempt to provide a secure foundation for mathematics, that is, a secure set of symbols and rules that all statements can be defined in, a proof that all true statements can be proven in this system, the system to be consistent, and there to be a single procedure to follow for deciding the truth or falsity of any mathematical statement in general. Kurt Gödel published his first and second incompleteness theorems in his 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems I, which were largely seen as ending any hopes for this project, casting doubt on the consistency of any such system, and definitively removing its completeness. Turing, in his 1937 paper On Computable Numbers, with an Application to the Entscheidungsproblem removed the last remaining remnant of the project, by showing that there is no general algorithm for finding the truth or falsity of any given mathematical statement.