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

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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  .

Examples

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Σ000 formulas and concepts

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  • 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

Δ1-formulas and concepts

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Σ1-formulas and concepts

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  • 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

Π1-formulas and concepts

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Δ2-formulas and concepts

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Σ2-formulas and concepts

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Π2-formulas and concepts

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Δ3-formulas and concepts

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Σ3-formulas and concepts

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Π3-formulas and concepts

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Σ4-formulas and concepts

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Properties

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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

See also

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References

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  • 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

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  1. ^ a b Walicki, Michal (2012). Mathematical Logic, p. 225. World Scientific Publishing Co. Pte. Ltd. ISBN 9789814343862
  2. ^ a b 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 b A. Lévy, 'A hierarchy of formulas in set theory' (1965), second edition
  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. ^ a b c d e f g h i j Jon Barwise, Admissible Sets and Structures. Perspectives in Mathematical Logic (1975)
  8. ^ a b c d e f D. Monk 2011, Graduate Set Theory (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. ^ F. R. Drake, Set Theory: An Introduction to Large Cardinals (p.127). Accessed 4 October 2024.
  13. ^ a b c 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