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 a similar classification 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 set membership predicates, respectively.
The first level of the Lévy hierarchy is defined as containing only formulas with no unbounded quantifiers and is denoted by. The next levels are given by finding a formula in prenex normal form that is provably equivalent over ZFC, and counting the number of alternations of quantifiers:p. 184
A formula is called:
  • if is equivalent to in ZFC, where is
  • if is equivalent to in ZFC, where is
  • If a formula is both and form, it is called.
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.
Lévy's original notation was due to the provable logical equivalence, strictly speaking the above levels should be referred to as to specify the theory in which the equivalence is carried out, however it is usually clear from context.pp. 441–442 Pohlers has defined in particular semantically, in which a formula is " in a structure ".
The Lévy hierarchy is sometimes defined for other theories S. In this case and by themselves refer only to formulas that start with a sequence of quantifiers with at most i−1 alternations, and and refer to formulas equivalent to and formulas in the language of the theory S. So strictly speaking the levels and of the Lévy hierarchy for ZFC defined above should be denoted by and .

Examples

Σ0=Π0=Δ0 formulas and concepts

Let. The Lévy hierarchy has the following properties:p. 184
  • If is, then is.
  • If is, then is.
  • If and are, then,,,, and are all.
  • If and are, then,,,, and are all.
  • If is and is, then is.
  • If is and is, then is.
Devlin p. 29