properties of ranks of sets
A set $A$ is said to be grounded, if $A\subseteq {V}_{\alpha}$ in the cumulative hierarchy for some ordinal^{} $\alpha $. The smallest such $\alpha $ such that $A\subseteq {V}_{\alpha}$ is called the rank of $A$, and is denoted by $\rho (A)$.
In this entry, we list derive some basic properties of groundedness and ranks of sets. Proofs of these properties require an understanding of some of the basic properties of ordinals.

1.
$\mathrm{\varnothing}$ is grounded, whose rank is itself. This is obvious.

2.
If $A$ is grounded, so is every $x\in A$, and $$.
Proof.
$A\subseteq {V}_{\rho (A)}$, so $x\in {V}_{\rho (A)}$, which means $x\subseteq {V}_{\beta}$ for some $$. This shows that $x$ is grounded. Then $\rho (x)\le \beta $, and hence $$. ∎

3.
If every $x\in A$ is grounded, so is $A$, and $\rho (A)=sup\{\rho {(x)}^{+}\mid x\in A\}$.
Proof.
Let $B=\{\rho {(x)}^{+}\mid x\in A\}$. Then $B$ is a set of ordinals, so that $\beta :=\bigcup B=supB$ is an ordinal. Since each $x\in {V}_{\rho {(x)}^{+}}$, we have $x\in {V}_{\beta}$. So $A\subseteq {V}_{\beta}$, showing that $A$ is grounded. If $$, then for some $x\in A$, $$, which means $x\notin {V}_{\alpha}$, and therefore $A\u2288{V}_{\alpha}$. This shows that $\rho (A)=\beta $. ∎

4.
If $A$ is grounded, so is $\{A\}$, and $\rho (\{A\})=\rho {(A)}^{+}$. This is a direct consequence of the previous result.

5.
If $A,B$ are grounded, so is $A\cup B$, and $\rho (A\cup B)=\mathrm{max}(\rho (A),\rho (B))$.
Proof.
Since $A,B$ are grounded, every element of $A\cup B$ is grounded by property 2, so that $A\cup B$ is also grounded by property 3. Then $\rho (A\cup B)=sup\{\rho {(x)}^{+}\mid x\in A\cup B\}=\mathrm{max}(sup\{\rho {(x)}^{+}\mid x\in A\},sup\{\rho {(x)}^{+}\mid x\in B\})=\mathrm{max}(\rho (A),\rho (B))$. ∎

6.
If $A$ is grounded, so is $B\subseteq A$, and $\rho (B)\le \rho (A)$.
Proof.
Every element of $B$, as an element of the grounded set $A$, is grounded, and therefore $B$ is grounded. So $\rho (B)=sup\{\rho {(x)}^{+}\mid x\in B\}\le sup\{\rho {(x)}^{+}\mid x\in A\}=\rho (A)$. Since $\rho (B)$ and $\rho (A)$ are both ordinals, $\rho (B)\le \rho (A)$. ∎

7.
If $A$ is grounded, so is $P(A)$, and $\rho (P(A))=\rho {(A)}^{+}$.
Proof.
Every subset of $A$ is grounded, since $A$ is by property 6. So $P(A)$ is grounded. Furthermore, $P(A)=sup\{\rho {(x)}^{+}\mid x\in P(A)\}$. Since $\rho (B)\le \rho (A)$ for any $B\in P(A)$, and $A\in P(A)$, we have $P(A)=\rho {(A)}^{+}$ as a result. ∎

8.
If $A$ is grounded, so is $\bigcup A$, and $\rho (\bigcup A)=sup\{\rho (x)\mid x\in A\}$.
Proof.
Since $A$ is grounded, every $x\in A$ is grounded. Let $B=\{\rho (x)\mid x\in A\}$. Then $\beta :=\bigcup B=supB$ is an ordinal. Since $\rho (x)\le \beta $, ${V}_{\rho (x)}={V}_{\beta}$ or ${V}_{\rho (x)}\in {V}_{\beta}$. In either case, ${V}_{\rho (x)}\subseteq {V}_{\beta}$, since ${V}_{\alpha}$ is a transitive set for any ordinal $\alpha $. Since $x\subseteq {V}_{\rho (x)}$, $x\subseteq {V}_{\beta}$ for every $x\in A$. This means $\bigcup A\subseteq {V}_{\beta}$, showing that $\bigcup A$ is grounded. If $$, then $$ for some $\rho (x)\le \beta $, which means $x\u2288{V}_{\alpha}$, or $\bigcup A\u2288{V}_{\alpha}$ as a result. Therefore $\rho (\bigcup A)=\beta $. ∎

9.
Every ordinal is grounded, whose rank is itself.
Proof.
If $\alpha =0$, then apply property 1. If $\alpha $ is a successor ordinal, apply properties 4 and 5, so that $\rho (\alpha )=\rho ({\beta}^{+})=\rho (\beta \cup \{\beta \})=\mathrm{max}(\rho (\beta ),\rho (\{\beta \}))=\mathrm{max}(\rho (\beta ),\rho {(\beta )}^{+})=\rho {(\beta )}^{+}$. If $\alpha $ is a limit ordinal, then apply property 8 and transfinite induction^{}, so that $$. ∎
References
 1 H. Enderton, Elements of Set Theory^{}, Academic Press, Orlando, FL (1977).
 2 A. Levy, Basic Set Theory, Dover Publications Inc., (2002).
Title  properties of ranks of sets 

Canonical name  PropertiesOfRanksOfSets 
Date of creation  20130322 18:50:31 
Last modified on  20130322 18:50:31 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Derivation 
Classification  msc 03E99 
Defines  grounded 
Defines  grounded set 