In this article we are going to explore *Lévy hierarchy*, a very relevant topic today that has aroused great interest in different sectors. *Lévy hierarchy* is a concept that has been the subject of debate and analysis in recent years, and its impact on society has been significant. Since its emergence, *Lévy hierarchy* has generated conflicting opinions and has been a source of reflection for experts and scholars in the field. Throughout this article, we will examine in depth the different aspects related to *Lévy hierarchy*, from its origin to its evolution over time, also addressing its implications and its influence in the corresponding field.

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.

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 .^{[1]} The next levels are given by finding a formula in prenex normal form which is provably equivalent over ZFC, and counting the number of changes of quantifiers:^{[2]}^{p. 184}

A formula is called:^{[1]}^{[3]}

- if is equivalent to in ZFC, where is
- if is equivalent to in ZFC, where is
- If a formula has both a form and a 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.^{[citation needed]}

Lévy's original notation was (resp. ) due to the provable logical equivalence,^{[4]} strictly speaking the above levels should be referred to as (resp. ) to specify the theory in which the equivalence is carried out, however it is usually clear from context.^{[5]}^{pp. 441–442} Pohlers has defined in particular semantically, in which a formula is " in a structure ".^{[6]}

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,^{[citation needed]} 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 .

- x = {y, z}
^{[7]}^{p. 14} - x ⊆ y
^{[8]} *x*is a transitive set^{[8]}*x*is an ordinal,*x*is a limit ordinal,*x*is a successor ordinal^{[8]}*x*is a finite ordinal^{[8]}- The first infinite ordinal ω
^{[8]} *x*is an ordered pair. The first entry of the ordered pair*x*is*a*. The second entry of the ordered pair*x*is*b*^{[7]}^{p. 14}*f*is a function.*x*is the domain/range of the function*f*.*y*is the value of*f*on*x*^{[7]}^{p. 14}- The Cartesian product of two sets.
*x*is the union of*y*^{[8]}*x*is a member of the*α*th level of Godel's L^{[9]}*R*is a relation with domain/range/field*a*^{[7]}^{p. 14}

*x*is a well-founded relation on*y*^{[10]}*x*is finite^{[4]}^{p.15}- Ordinal addition and multiplication and exponentiation
^{[11]} - The rank (with respect to Gödel's constructible universe) of a set
^{[7]}^{p. 61} - The transitive closure of a set.
- The specifiability relation Sp(A) for a set A.
^{[12]}

*x*is countable.- |
*X*|≤|*Y*|, |*X*|=|*Y*|. *x*is constructible.*g*is the restriction of the function*f*to*a*^{[7]}^{p. 23}*g*is the image of*f*on*a*^{[7]}^{p. 23}*b*is the successor ordinal of*a*^{[7]}^{p. 23}- rank(
*x*)^{[7]}^{p. 29} - The Mostowski collapse of
^{[7]}^{p. 29}

*x*is a cardinal*x*is a regular cardinal*x*is a limit cardinal*x*is an inaccessible cardinal.*x*is the powerset of*y*

*κ*is*γ*-supercompact

- the continuum hypothesis
- there exists an inaccessible cardinal
- there exists a measurable cardinal
*κ*is an*n*-huge cardinal

- The axiom of choice
^{[13]} - The generalized continuum hypothesis
^{[13]} - The axiom of constructibility:
*V*=*L*^{[13]}

- there exists a supercompact cardinal

*κ*is an extendible cardinal

- there exists an extendible cardinal

Let . The Lévy hierarchy has the following properties:^{[2]}^{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

- Devlin, Keith J. (1984).
*Constructibility*. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. pp. 27–30. Zbl 0542.03029. - Jech, Thomas (2003).
*Set Theory*. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. p. 183. ISBN 978-3-540-44085-7. Zbl 1007.03002. - Kanamori, Akihiro (2006). "Levy and set theory".
*Annals of Pure and Applied Logic*.**140**(1–3): 233–252. doi:10.1016/j.apal.2005.09.009. Zbl 1089.03004. - Levy, Azriel (1965).
*A hierarchy of formulas in set theory*. Mem. Am. Math. Soc. Vol. 57. Zbl 0202.30502.

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