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.

Definitions

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 \Delta _0=\Sigma_0=\Pi_0.[1] 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 A is called:[1]

\Sigma _{i+1} if A is equivalent to \exists x _1 ... \exists x _n B in ZFC, where B is \Pi _i

\Pi _{i+1} if A is equivalent to \forall x _1 ... \forall x _n B in ZFC, where B is \Sigma _i

If a formula is both \Sigma _i and \Pi _i, it is called \Delta _i. As a formula might have several different equivalent formulas in Prenex normal form, it might belong to several different levels of the hierarchy. In this case, the lowest possible level is the level of the formula.

The Lévy hierarchy is sometimes defined for other theories S. In this case \Sigma _i and \Pi _i by themselves refer only to formulas that start with a sequence of quantifiers with at most i−1 alternations, and \Sigma _i^S and \Pi _i^S refer to formulas equivalent to \Sigma _i and \Pi _i formulas in the theory S. So strictly speaking the levels \Sigma _i and \Pi _i of the Lévy hierarchy for ZFC defined above should be denoted by \Sigma^{ZFC} _i and \Pi^{ZFC} _i.

Examples

Σ000 formulas and concepts

Δ1-formulas and concepts

Σ1-formulas and concepts

Π1-formulas and concepts

Δ2-formulas and concepts

Σ2-formulas and concepts

Π2-formulas and concepts

Δ3-formulas and concepts


Σ3-formulas and concepts

Π3-formulas and concepts

Σ4-formulas and concepts

Properties

Jech p. 184 Devlin p. 29

See also

References

  1. 1 2 Walicki, Michal (2012). Mathematical Logic, p. 225. World Scientific Publishing Co. Pte. Ltd. ISBN 9789814343862


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