Lévy hierarchy

In set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy of formulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language of set theory. This is analogous to the arithmetical hierarchy which provides the classifications but for sentences of the language of arithmetic.

In the language of set theory, atomic formulas are of the form x = y or x ∈ y, standing for equality and respectively set membership predicates.

The first level of the Levy hierarchy is defined as containing only formulas with no unbounded quantifiers, and is denoted by ${\displaystyle \Delta _{0}=\Sigma _{0}=\Pi _{0}}$. The next levels are given by finding an equivalent formula in Prenex normal form, and counting the number of changes of quantifiers:

In the theory ZFC, a formula ${\displaystyle A}$ is called:

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