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 .[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.

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, 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

Σ000 formulas and concepts

Δ1-formulas and concepts

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 [7]p. 29

Π1-formulas and concepts

Δ2-formulas and concepts

Σ2-formulas and concepts

Π2-formulas and concepts

Δ3-formulas and concepts

Σ3-formulas and concepts

Π3-formulas and concepts

Σ4-formulas and concepts

Properties

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

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. 1 2 Walicki, Michal (2012). Mathematical Logic, p. 225. World Scientific Publishing Co. Pte. Ltd. ISBN 9789814343862
  2. 1 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. 1 2 3 4 5 6 7 8 9 10 Jon Barwise, Admissible Sets and Structures. Perspectives in Mathematical Logic (1975)
  8. 1 2 3 4 5 6 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. 1 2 3 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
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