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Axiom Set Zermelo Fraenkel Axioms Set Theory Ordinal Number Von Neumann Ordinal Axiom Of Foundation Transitive Transfinite Recursion

1	Set Theory
1	Set
1	Axiom
1	Ordinal Number
1	Zermelo Fraenkel Axioms
2	Von Neumann Ordinal
3	Axiom Of Foundation
5	Transfinite Recursion
6	Transitive

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Cumulative Hierarchy

The cumulative hierarchy of sets is defined by transfinite recursion as follows: we define $V_0=\varnothing$ and for each ordinal $\alpha$ we define $V_{\alpha+1}=\mathcal{P}(V_\alpha)$ and for each limit ordinal $\delta$ we define $V_\delta=\bigcup_{\alpha\in\delta}V_\alpha$ .

Every set is a subset of $V_\alpha$ for some ordinal $\alpha$ , and the least such $\alpha$ 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 $\alpha$ , the set $V_\alpha$ is the set of all sets of rank less than $\alpha$ , and $V_{\alpha+1}\setminus V_\alpha$ is the set of all sets of rank $\alpha$ .

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 $V_\alpha$ 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 $V_\alpha$ is a transitive set. Note that , and , but for $\alpha>2$ the set $V_\alpha$ is never an ordinal, because it has the element $\{1\}$ , which is not an ordinal.