Kolmogorov's three-series theorem

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 (LST). This is analogous to the arithmetical hierarchy which provides the classifications but for sentences of the language of arithmetic.

In LST, atomic formulas are of the form x = y or x ∈ y, standing for equality and respectively set membership predicates. (It is possible to even define equality within ZF by slightly different formulation of one of the axioms, although that issue has no impact on the topic of this article.)

The first level of the Levy hierarchy is defined as containing only formulas in which all quantifiers are bounded, meaning only of the form $\forall x\in y$ and $\exists x\in y$. This level of the Levy hierarchy is denoted by any and all of Δ₀, Σ₀, Π₀. Then Σ_n+1 is defined as

Examples

Σ₀-formulas:

• x = {y, z}
• x ⊆ y
• x is a transitive set
• x is an ordinal

Σ₁-formulas:

• x is countable
• x is finite

Π₁-formulas:

• x is a cardinal
• x is a regular cardinal
• x is a limit cardinal

Δ₁-formulas:

• x is a well-founded relation on y

Σ₂-formulas:

• the Continuum Hypothesis (and its negation)
• there exists an inaccessible cardinal
• there exists a measurable cardinal
• V ≠ L

Properties

Jech p. 184 Devlin p. 29

See also

• arithmetic hierarchy
• Absoluteness

References

• Joan Bagaria, A gentle introduction to the theory of large cardinals
• Devlin, Constructibility pp. 27–30
• A. Lévy. A hierarchy of formulas in set theory. Mem. Am. Math. Soc., 57 (1965)

76 pp.

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