Application To Computable Functions
Goodstein's theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total. The Goodstein sequence of a number can be effectively enumerated by a Turing machine; thus the function which maps n to the number of steps required for the Goodstein sequence of n to terminate is computable by a particular Turing machine. This machine merely enumerates the Goodstein sequence of n and, when the sequence reaches 0, returns the length of the sequence. Because every Goodstein sequence eventually terminates, this function is total. But because Peano arithmetic does not prove that every Goodstein sequence terminates, Peano arithmetic does not prove that this Turing machine computes a total function.
Read more about this topic: Goodstein's Theorem
Famous quotes containing the words application to, application and/or functions:
“If you would be a favourite of your king, address yourself to his weaknesses. An application to his reason will seldom prove very successful.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“If photography is allowed to stand in for art in some of its functions it will soon supplant or corrupt it completely thanks to the natural support it will find in the stupidity of the multitude. It must return to its real task, which is to be the servant of the sciences and the arts, but the very humble servant, like printing and shorthand which have neither created nor supplanted literature.”
—Charles Baudelaire (1821–1867)