The Arithmetical Hierarchy of Sets of Natural Numbers
A set X of natural numbers is defined by formula φ in the language of Peano arithmetic if the elements of X are exactly the numbers that satisfy φ. That is, for all natural numbers n,
where is the numeral in the language of arithmetic corresponding to . A set is definable in first order arithmetic if it is defined by some formula in the language of Peano arithmetic.
Each set X of natural numbers that is definable in first order arithmetic is assigned classifications of the form, and, where is a natural number, as follows. If X is definable by a formula then X is assigned the classification . If X is definable by a formula then X is assigned the classification . If X is both and then is assigned the additional classification .
Note that it rarely makes sense to speak of formulas; the first quantifier of a formula is either existential or universal. So a set is not defined by a formula; rather, there are both and formulas that define the set.
A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of the natural numbers. Instead of formulas with one free variable, formulas with k free number variables are used to define the arithmetical hierarchy on sets of k-tuples of natural numbers.
Read more about this topic: Arithmetical Hierarchy
Famous quotes containing the words hierarchy, sets, natural and/or numbers:
“In a hierarchy every employee tends to rise to his level of incompetence.”
—Laurence J. Peter (1919–1990)
“The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveller.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Saints are simply men & women who have fulfilled their natural obligation which is to approach God.”
—Evelyn Waugh (1903–1966)
“I’m not even thinking straight any more. Numbers buzz in my head like wasps.”
—Kurt Neumann (1906–1958)