Cumulative hierarchy ($V_{\alpha }$)

The cumulative hierarchy is defined by $V_0=\varnothing$, $V_{\alpha +1}=\mathcal{P}(V_{\alpha })$, and $V_{\lambda }={\bigcup }_{\alpha <\lambda }{V}_{\alpha }$ at limit ordinals.

The transitive closure $\mathrm{TC}(x)$ is the least transitive set containing every member of $x$. It is obtained by taking the union of all finite iterates of the union operation starting from $x$.

For an infinite cardinal number $\kappa$, $H_{\kappa }$ consists of the sets $x$ for which $|\mathrm{tc}(\{x\})|<\kappa$, where $\mathrm{tc}$ is transitive closure. The entire transitive membership ancestry is bounded, not only the size of $x$. These collections are transitive sets. At $\kappa ={\omega }_1$ the elements are the hereditarily countable sets.

The hereditarily small set structure $H_{{\omega }_2}$ fails the Axiom of power set at ${\omega }_1$. Every subset of ${\omega }_1$ belongs to $H_{{\omega }_2}$, but their full power set has cardinality at least ${\aleph }_2$ by the Cantor theorem, so cannot itself be hereditarily smaller than ${\aleph }_2$.

Let $z_0=\omega$ and $z_{n+1}=\mathcal{P}(z_n)$. Call $x$ small when it injects into some $z_n$, and let $\mathbf{HS}$ contain exactly the sets all of whose hereditary members are small. Then every $z_n$ and the set $y=\{z_n:n\in \omega \}$ belong to $\mathbf{HS}$, but $\bigcup y$ does not. Consequently $(\mathbf{HS},\in )$ fails the Axiom of union.

A set is hereditarily countable when its transitive closure is countable. These sets form the transitive set $H_{{\omega }_1}$ and all have rank below ${\omega }_1$.

A set $x$ is reasonable when every member of $\mathrm{TC}(\{x\})$ is countable. In ZFC this is equivalent to $x$ being hereditarily countable.

The class of hereditarily finite sets is the setEquivalently, these are the sets with finite transitive closure, or the sets contained in a finite transitive set.

A finite von Neumann ordinal is obtained from $0=\varnothing$ by finitely many applications of the successor operation $S(x)=x\cup \{x\}$. Equivalently, it is an ordinal every nonempty subset of which has a greatest element. This characterization defines the individual natural numbers without quantifying over all inductive sets.