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.

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 ${\displaystyle \Delta _{0}=\Sigma _{0}=\Pi _{0}}$.^[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 ${\displaystyle A}$ is called:^[1][3]

• ${\displaystyle \Sigma _{i+1}}$ if ${\displaystyle A}$ is equivalent to ${\displaystyle \exists x_{1}...\exists x_{n}B}$ in ZFC, where ${\displaystyle B}$ is ${\displaystyle \Pi _{i}}$
• ${\displaystyle \Pi _{i+1}}$ if ${\displaystyle A}$ is equivalent to ${\displaystyle \forall x_{1}...\forall x_{n}B}$ in ZFC, where ${\displaystyle B}$ is ${\displaystyle \Sigma _{i}}$
• If a formula has both a ${\displaystyle \Sigma _{i}}$ form and a ${\displaystyle \Pi _{i}}$ form, it is called ${\displaystyle \Delta _{i}}$.

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.^{[citation needed]}

Lévy's original notation was ${\displaystyle \Sigma _{i}^{\mathsf {ZFC}}}$ (resp. ${\displaystyle \Pi _{i}^{\mathsf {ZFC}}}$) due to the provable logical equivalence,^[4] strictly speaking the above levels should be referred to as ${\displaystyle \Sigma _{i}^{\mathsf {ZFC}}}$ (resp. ${\displaystyle \Pi _{i}^{\mathsf {ZFC}}}$) 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 ${\displaystyle \Delta _{1}}$ in particular semantically, in which a formula is "${\displaystyle \Delta _{1}}$ in a structure ${\displaystyle M}$".^[6]

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

• 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 αth 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 ${\displaystyle (x,\in )}$ ^[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 ${\displaystyle n\geq 1}$. The Lévy hierarchy has the following properties:^[2]p. 184

• If ${\displaystyle \phi }$ is ${\displaystyle \Sigma _{n}}$, then ${\displaystyle \lnot \phi }$ is ${\displaystyle \Pi _{n}}$.
• If ${\displaystyle \phi }$ is ${\displaystyle \Pi _{n}}$, then ${\displaystyle \lnot \phi }$ is ${\displaystyle \Sigma _{n}}$.
• If ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are ${\displaystyle \Sigma _{n}}$, then ${\displaystyle \exists x\phi }$, ${\displaystyle \phi \land \psi }$, ${\displaystyle \phi \lor \psi }$, ${\displaystyle \exists (x\in z)\phi }$, and ${\displaystyle \forall (x\in z)\phi }$ are all ${\displaystyle \Sigma _{n}}$.
• If ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are ${\displaystyle \Pi _{n}}$, then ${\displaystyle \forall x\phi }$, ${\displaystyle \phi \land \psi }$, ${\displaystyle \phi \lor \psi }$, ${\displaystyle \exists (x\in z)\phi }$, and ${\displaystyle \forall (x\in z)\phi }$ are all ${\displaystyle \Pi _{n}}$.
• If ${\displaystyle \phi }$ is ${\displaystyle \Sigma _{n}}$ and ${\displaystyle \psi }$ is ${\displaystyle \Pi _{n}}$, then ${\displaystyle \phi \implies \psi }$ is ${\displaystyle \Pi _{n}}$.
• If ${\displaystyle \phi }$ is ${\displaystyle \Pi _{n}}$ and ${\displaystyle \psi }$ is ${\displaystyle \Sigma _{n}}$, then ${\displaystyle \phi \implies \psi }$ is ${\displaystyle \Sigma _{n}}$.

Devlin p. 29

• Arithmetic hierarchy
• Absoluteness

• 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.