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.
|Date of creation||2013-03-22 16:18:43|
|Last modified on||2013-03-22 16:18:43|
|Last modified by||yark (2760)|
|Defines||rank of a set|