The cumulative hierarchy is defined inductively as follows:

$V_{0} = \emptyset$

$V_{\alpha + 1} = \mathcal{P}({V_{\alpha}})$, the powerset of $V_{\alpha}$

$V_{\delta} = \bigcup_{\alpha < \delta} V_{\alpha}$ if $\delta$ is a limit ordinal.

The $ZFC$ axioms essentially say that the set theoretic universe $V$ is the union of the $V_{\alpha}$, where $\alpha$ runs through the ordinals. It turns out that a collection of sets $X$ is a set if and only there exists an ordinal $\alpha$ such that $X \subseteq V_{\alpha}$. Thus $X$ is a set if and only if $X$ has finished being created before the entire universe is created.

The cumulative hierarchy is defined inductively as follows:

$V_{0} = \emptyset$

$V_{\alpha + 1} = \mathcal{P}({V_{\alpha}})$, the powerset of $V_{\alpha}$

$V_{\delta} = \bigcup_{\alpha < \delta} V_{\alpha}$ if $\delta$ is a limit ordinal.

The $ZFC$ axioms essentially say that the set theoretic universe $V$ is the union of the $V_{\alpha}$, where $\alpha$ runs through the ordinals. It turns out that a collection of sets $X$ is a set if and only there exists an ordinal $\alpha$ such that $X \subseteq V_{\alpha}$. Thus $X$ is a set if and only if $X$ has ``finished being created'' before the entire universe is created.

The cumulative hierarchy is defined inductively as follows:

$V_{0} = \emptyset$

$V_{\alpha + 1} = \mathcal{P}({V_{\alpha}})$

$V_{\delta} = \bigcup_{\alpha < \delta} V_{\alpha}$ if $\delta$ is a limit ordinal.

The $ZFC$ axioms essentially say that the set theoretic universe $V$ is the union of the $V_{\alpha}$, where $\alpha$ runs through the ordinals. It turns out that a collection of sets $X$ is a set if and only there exists an ordinal $\alpha$ such that $X \subseteq V_{\alpha}$. Thus $X$ is a set if and only if $X$ has finished being created before the entire universe is created.

The cumulative hierarchy is defined inductively as follows:

$V_{0} = \emptyset$

$V_{\alpha + 1} = \mathcal{P}({V_{\alpha}})$, the powerset of $V_{\alpha}$

$V_{\delta} = \bigcup_{\alpha < \delta} V_{\alpha}$ if $\delta$ is a limit ordinal.

The $ZFC$ axioms essentially say that the set theoretic universe $V$ is the union of the $V_{\alpha}$, where $\alpha$ runs through the ordinals. It turns out that a collection of sets $X$ is a set if and only there exists an ordinal $\alpha$ such that $X \subseteq V_{\alpha}$. Thus $X$ is a set if and only if $X$ has finished being created before the entire universe is created.