Kolmogorov's three-series theorem: Difference between revisions

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{{Expert-subject|Set theory|date=December 2012}}
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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 formula]]s are of the form x = y or x ∈ y, standing for [[Equality (mathematics)|equality]] and respectively [[set membership]] predicates. (It is possible to even [[First-order_logic#Defining_equality_within_a_theory|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_quantifier#Bounded_quantifiers_in_set_theory|bounded]], meaning only of the form <math>\forall x \in y</math> and <math>\exists x \in y</math>. This level of the Levy hierarchy is denoted by any and all of Δ<sub>0</sub>, Σ<sub>0</sub>, Π<sub>0</sub>. Then Σ<sub>n+1</sub> is defined as
 
== Examples ==
Σ<sub>0</sub>-formulas:
* x = {y, z}
* x ⊆ y
* ''x'' is a [[transitive set]]
* ''x'' is an [[Ordinal number|ordinal]]
 
Σ<sub>1</sub>-formulas:
* ''x'' is [[Countable set|countable]]
* ''x'' is finite
 
Π<sub>1</sub>-formulas:
* ''x'' is a [[cardinal number|cardinal]]
* ''x'' is a [[regular cardinal]]
* ''x'' is a [[limit cardinal]]
 
Δ<sub>1</sub>-formulas:
* ''x'' is a [[well-founded relation]] on ''y''
 
Σ<sub>2</sub>-formulas:
* the [[Continuum Hypothesis]] (and its negation)
* there exists an [[inaccessible cardinal]]
* there exists a [[measurable cardinal]]
* ''[[Von Neumann universe|V]]'' ≠ ''[[Constructible universe|L]]''
 
== Properties ==
Jech p.&nbsp;184
Devlin p.&nbsp;29
 
== See also ==
* [[arithmetic hierarchy]]
* [[Absoluteness]]
 
== References ==
* Joan Bagaria, [http://www.imub.ub.es/hocard11/Bagaria_slides.pdf A gentle introduction to the theory of large cardinals]
* Devlin, Constructibility pp.&nbsp;27–30
* A. Lévy. A hierarchy of formulas in set theory. Mem. Am. Math. Soc., 57 (1965)
76 pp.
 
{{DEFAULTSORT:Levy hierarchy}}
[[Category:Mathematical logic]]
[[Category:Set theory]]
 
 
{{settheory-stub}}

Latest revision as of 17:00, 10 December 2014

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