# 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. 1.

$\varnothing$ is grounded, whose rank is itself. This is obvious.

2. 2.

If $A$ is grounded, so is every $x\in A$, and $\rho(x)<\rho(A)$.

###### Proof.

$A\subseteq V_{\rho(A)}$, so $x\in V_{\rho(A)}$, which means $x\subseteq V_{\beta}$ for some $\beta<\rho(A)$. This shows that $x$ is grounded. Then $\rho(x)\leq\beta$, and hence $\rho(x)<\rho(A)$. ∎

3. 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 $\beta:=\bigcup B=\sup B$ 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 $\alpha<\beta$, then for some $x\in A$, $\alpha<\rho(x)^{+}$, which means $x\notin V_{\alpha}$, and therefore $A\nsubseteq V_{\alpha}$. This shows that $\rho(A)=\beta$. ∎

4. 4.

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

5. 5.

If $A,B$ are grounded, so is $A\cup B$, and $\rho(A\cup B)=\max(\rho(A),\rho(B))$.

###### 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 $\rho(A\cup B)=\sup\{\rho(x)^{+}\mid x\in A\cup B\}=\max(\sup\{\rho(x)^{+}\mid x% \in A\},\sup\{\rho(x)^{+}\mid x\in B\})=\max(\rho(A),\rho(B))$. ∎

6. 6.

If $A$ is grounded, so is $B\subseteq A$, and $\rho(B)\leq\rho(A)$.

###### Proof.

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

7. 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 $\rho(B)\leq\rho(A)$ for any $B\in P(A)$, and $A\in P(A)$, we have $P(A)=\rho(A)^{+}$ as a result. ∎

8. 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 $\beta:=\bigcup B=\sup B$ is an ordinal. Since $\rho(x)\leq\beta$, $V_{\rho(x)}=V_{\beta}$ 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 $\alpha<\beta$, then $\alpha<\rho(x)$ for some $\rho(x)\leq\beta$, which means $x\nsubseteq V_{\alpha}$, or $\bigcup A\nsubseteq V_{\alpha}$ as a result. Therefore $\rho(\bigcup A)=\beta$. ∎

9. 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 $\rho(\alpha)=\rho(\beta^{+})=\rho(\beta\cup\{\beta\})=\max(\rho(\beta),\rho(\{% \beta\}))=\max(\rho(\beta),\rho(\beta)^{+})=\rho(\beta)^{+}$. If $\alpha$ is a limit ordinal, then apply property 8 and transfinite induction, so that $\rho(\alpha)=\rho(\bigcup\alpha)=\sup\{\rho(\beta)\mid\beta<\alpha\}=\sup\{% \beta\mid\beta<\alpha\}=\alpha$. ∎

## 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 PropertiesOfRanksOfSets 2013-03-22 18:50:31 2013-03-22 18:50:31 CWoo (3771) CWoo (3771) 8 CWoo (3771) Derivation msc 03E99 grounded grounded set