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
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