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)
“bars of that strange speech
In which each sound sets out to seek each other,
Murders its own father, marries its own mother,
And ends as one grand transcendental vowel.”
—Randall Jarrell (1914–1965)
“May she be granted beauty and yet not
Beauty to make a stranger’s eye distraught,
Or hers before a looking-glass, for such,
Being made beautiful overmuch,
Consider beauty a sufficient end,
Lose natural kindness”
—William Butler Yeats (1865–1939)
“He bundles every forkful in its place,
And tags and numbers it for future reference,
So he can find and easily dislodge it
In the unloading. Silas does that well.
He takes it out in bunches like birds’ nests.”
—Robert Frost (1874–1963)