Cumulative hierarchy

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, a cumulative hierarchy is a family of <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> 
 
 
 
 
 W
 
 α
 
 
 
 
 {\displaystyle W_{\alpha }}
 
 indexed by <a href="/facts/Ordinal_number/GelFLjJt">ordinals</a> 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 such that

<ul><li>
 
 
 
 
 W
 
 α
 
 
 ⊆
 
 W
 
 α
 +
 1
 
 
 
 
 {\displaystyle W_{\alpha }\subseteq W_{\alpha +1}}
 
</li>
<li>If 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is a <a href="/facts/Limit_ordinal/jPCCoPuP">limit ordinal</a>, then 
 
 
 
 
 W
 
 λ
 
 
 =
 
 ⋃
 
 α
 <
 λ
 
 
 
 W
 
 α
 
 
 
 
 {\textstyle W_{\lambda }=\bigcup _{\alpha <\lambda }W_{\alpha }}
 
</li></ul>
Some authors additionally require that 
 
 
 
 
 W
 
 α
 +
 1
 
 
 ⊆
 
 
 P
 
 
 (
 
 W
 
 α
 
 
 )
 
 
 {\displaystyle W_{\alpha +1}\subseteq {\mathcal {P}}(W_{\alpha })}
 
.
The <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> 
 
 
 
 W
 =
 
 ⋃
 
 α
 ∈
 
 O
 n
 
 
 
 
 W
 
 α
 
 
 
 
 {\textstyle W=\bigcup _{\alpha \in \mathrm {On} }W_{\alpha }}
 
 of the sets of a cumulative hierarchy is often used as a model of set theory.
The phrase "the cumulative hierarchy" usually refers to the <a href="/facts/Von_Neumann_universe/EhkJAxXc">von Neumann universe</a>, which has 
 
 
 
 
 W
 
 α
 +
 1
 
 
 =
 
 
 P
 
 
 (
 
 W
 
 α
 
 
 )
 
 
 {\displaystyle W_{\alpha +1}={\mathcal {P}}(W_{\alpha })}
 
.

Cumulative hierarchy open-in-new

Cumulative hierarchy