cumulative hierarchy
The cumulative hierarchy of setsis defined by transfinite recursion as follows:we define and for each ordinal we define and for each limit ordinal
we define.
Every set is a subset of for some ordinal ,and the least such 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 is the least ordinal greater than the rank of every element of .For each ordinal ,the set is the set of all sets of rank less than ,and is the set of all sets of rank .
Note that the previous paragraph makes use of the Axiom of Foundation:if this axiom fails,then there are sets that are not subsets of any and therefore have no rank.The previous paragraph also assumes that we are using a set theory
such as ZF,in which elements of sets are themselves sets.
Each is a transitive set.Note that , and ,but for the set is never an ordinal,because it has the element , 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 |