| Version 3 |
Version 2 |
| The \emph{cumulative hierarchy} of sets |
The \emph{cumulative hierarchy} of sets |
| is defined by transfinite recursion as follows: |
is defined by transfinite recursion as follows: |
| we define $V_0=\varnothing$ |
we define $V_0=\varnothing$ |
| and for each ordinal $\alpha$ we define $V_{\alpha+1}=\mathcal{P}(V_\alpha)$ |
and for each ordinal $\alpha$ we define $V_{\alpha+1}=\mathcal{P}(V_\alpha)$ |
| and for each limit ordinal $\delta$ we define |
and for each limit ordinal $\delta$ we define |
| $V_\delta=\bigcup_{\alpha\in\delta}V_\alpha$. |
$V_\delta=\bigcup_{\alpha\in\delta}V_\alpha$. |
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| Every set is a subset of $V_\alpha$ for some ordinal $\alpha$, |
Every set is a subset of $V_\alpha$ for some ordinal $\alpha$, |
| and the least such $\alpha$ is called the \emph{rank} of the set. |
and the least such $\alpha$ is called the \emph{rank} of the set. |
| It can be shown that the rank of an ordinal is itself, |
It can be shown that the rank of an ordinal is itself, |
| and in general the rank of a set $X$ |
and in general the rank of a set $X$ |
| is the least ordinal greater than the rank of every element of $X$. |
is the least ordinal greater than the rank of every element of $X$. |
| For each ordinal $\alpha$, |
For each ordinal $\alpha$, |
| the set $V_\alpha$ is the set of all sets of rank less than $\alpha$, |
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 $\alpha$. |
and $V_{\alpha+1}\setminus V_\alpha$ is the set of all sets of rank $\alpha$. |
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| Note that the previous paragraph makes use of the Axiom of Foundation: |
Note that the previous paragraph makes use of the Axiom of Foundation: |
| if this axiom fails, |
if this axiom fails, |
| then there are sets that are not subsets of any $V_\alpha$ |
then there are sets that are not subsets of any $V_\alpha$ |
| and therefore have no rank. |
and therefore have no rank. |
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| Each $V_\alpha$ is a transitive set. |
Each $V_\alpha$ is a transitive set. |
| Note that $V_0=0$, $V_1=1$ and $V_2=2$, |
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| but for $\alpha>2$ the set $V_\alpha$ is never an ordinal, |
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| because it has the element $\{1\}$, which is not an ordinal. |
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