cumulative hierarchy

The cumulative hierarchy of sets is defined by transfinite recursion as follows: we define V0= and for each ordinalMathworldPlanetmathPlanetmath α we define Vα+1=𝒫(Vα) and for each limit ordinalMathworldPlanetmath δ we define Vδ=αδVα.

Every set is a subset of Vα for some ordinal α, and the least such α is called the rank of the set. It can be shown that the rank of an ordinal is itself, and in general the rank of a set X is the least ordinal greater than the rank of every element of X. For each ordinal α, the set Vα is the set of all sets of rank less than α, and Vα+1Vα is the set of all sets of rank α.

Note that the previous paragraph makes use of the Axiom of FoundationMathworldPlanetmath: if this axiom fails, then there are sets that are not subsets of any Vα and therefore have no rank. The previous paragraph also assumes that we are using a set theoryMathworldPlanetmath such as ZF, in which elements of sets are themselves sets.

Each Vα is a transitive set. Note that V0=0, V1=1 and V2=2, but for α>2 the set Vα is never an ordinal, because it has the element {1}, which is not an ordinal.

Title cumulative hierarchy
Canonical name CumulativeHierarchy
Date of creation 2013-03-22 16:18:43
Last modified on 2013-03-22 16:18:43
Owner yark (2760)
Last modified by yark (2760)
Numerical id 8
Author yark (2760)
Entry type Definition
Classification msc 03E99
Synonym iterative hierarchy
Synonym Zermelo hierarchy
Related topic CriterionForASetToBeTransitive
Related topic ExampleOfUniverse
Defines rank
Defines rank of a set