Arithmetical hierarchy

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946).

The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.

The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines.

The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets.

The arithmetical hierarchy of formulas

The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted

Σ

n

0

{\displaystyle \Sigma _{n}^{0}}

and

Π

n

0

{\displaystyle \Pi _{n}^{0}}

for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain set parameters.

If a formula

ϕ

{\displaystyle \phi }

is logically equivalent to a formula in which all quantifiers are bounded then

ϕ

{\displaystyle \phi }

is assigned the classifications

Σ

0

0

{\displaystyle \Sigma _{0}^{0}}

and

Π

0

0

{\displaystyle \Pi _{0}^{0}}

.