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 ${\displaystyle \Delta _{0}=\Sigma _{0}=\Pi _{0}}$ .^[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 ${\displaystyle \Pi _{i}^{ZFC}}$ .

Examples

Σ₀=Π₀=Δ₀ formulas and concepts

• 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 countable 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

Δ₁-formulas and concepts

• x is a well-founded relation on y ^[10]
• x is finite.
• 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.

Σ₁-formulas and concepts

• 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

Π₁-formulas and concepts

• 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

Δ₂-formulas and concepts

• κ is γ-supercompact

Σ₂-formulas and concepts

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

Π₂-formulas and concepts

• The axiom of choice^[12]
• The generalized continuum hypothesis^[12]
• The axiom of constructibility: V = L^[12]

Δ₃-formulas and concepts

Σ₃-formulas and concepts

• there exists a supercompact cardinal

Π₃-formulas and concepts

• κ is an extendible cardinal

Σ₄-formulas and concepts

• there exists an extendible cardinal

Properties

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

See also

• Arithmetic hierarchy
• Absoluteness

References

• 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.

Citations

1. Walicki, Michal (2012). Mathematical Logic, p. 225. World Scientific Publishing Co. Pte. Ltd. ISBN 9789814343862
2. T. Jech, 'Set Theory: The Third Millennium Edition, revised and expanded'. Springer Monographs in Mathematics (2006). ISBN 3-540-44085-2.
3. J. Baeten, Filters and ultrafilters over definable subsets over admissible ordinals (1986). p.10
4. A. Lévy, 'A hierarchy of formulas in set theory' (1965)
5. K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of Symbolic Logic vol. 56, iss. 2 (1991), pp.439--457.
6. W. Pohlers, Proof Theory: The First Step into Impredicativity (2009) (p.245)
7. Jon Barwise, Admissible Sets and Structures. Perspectives in Mathematical Logic (1975)
8. D. Monk 2011, (pp.168--170). Archived 2011-12-06
9. W. A. R. Weiss, An Introduction to Set Theory (chapter 13). Accessed 2022-12-01
10. K. J. Williams, Minimum models of second-order set theories (2019, p.4). Accessed 2022 July 25.
11. F. R. Drake, Set Theory: An Introduction to Large Cardinals (p.83). Accessed 1 July 2022.
12. Azriel Lévy, "On the logical complexity of several axioms of set theory" (1971). Appearing in Axiomatic Set Theory: Proceedings of Symposia in Pure Mathematics, vol. 13 part 1, pp.219--230