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\begin{document}

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\section*{cumulative hierarchy}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_zfc}{In ZFC}\dotfill \pageref*{in_zfc} \linebreak
\noindent\hyperlink{in_algebraic_set_theory}{In algebraic set theory}\dotfill \pageref*{in_algebraic_set_theory} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[material set theory]], especially versions of set theory that accept the [[axiom of foundation]] that the membership relation $\in$ is an extensional (but not necessarily [[transitive relation|transitive]]) [[well-founded relation]], there is a picture of sets as forming a \emph{cumulative hierarchy}, hierarchically ordered by a rank function valued in the ordinals. The idea is that on Day 0 the empty set is born; on Day $n+1$ is born the power set of Day $n$. On limit Days all the sets born earlier are collected together into a single set which is their union. The cumulative hierarchy picture amounts to the assertion that every set belongs to some set produced by this iterative procedure.

\hypertarget{in_zfc}{}\subsection*{{In ZFC}}\label{in_zfc}

Suppose $V$ is a [[model]] of [[ZFC]]. Recall that in the theory ZFC, a set $x$ is defined to be \emph{transitive} if $b \in x$ implies $b \subset x$; this is the same as saying that if $a \in b \in x$ then $a \in x$. Note this doesn't mean that the relation $\in$ on the set consisting of $x$ and the elements of its transitive closure is itself [[transitive relation|transitive]]: the condition concerns only chains $a \in b \in c$ where $c = x$. However, if that relation \emph{is} transitive, then it is transitive, and extensional (by the extensionality axiom), and well-founded (by the axiom of foundation), and hence a well-order according to the argument \href{/nlab/show/well-order#wellorders_are_linear}{here}. In this case, $x$ is by definition an [[ordinal number]] (in the sense of von Neumann).

Letting $On(V)$ be the class of ordinals (= ordinal numbers), one may define with the help of the replacement axiom a function

\begin{displaymath}
R: On(V) \to V
\end{displaymath}
by transfinite induction as follows: $R(0) = 0$, while

\begin{itemize}%
\item $R(\alpha + 1) = P(R(\alpha))$ (where $P$ denotes power set),


\item $R(\beta) = \bigcup_{\alpha \lt \beta} R(\alpha)$ when $\beta$ is a limit ordinal.



\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
Each of the $R(\alpha)$ is transitive, and $\alpha \leq \beta$ implies $R(\alpha) \subseteq R(\beta)$.

\end{prop}
\begin{proof}
First, if $X$ is a transitive set, then so is $P(X)$. For suppose $A \in P(X)$. Then $A \subseteq X$, and so if $x \in A$, we have $x \in X$ and thus $x \subset X$ since $X$ is transitive, so that $x \in P(X)$. We have thus shown $A \subset P(X)$.

That each $R(\alpha)$ is transitive now follows by an easy induction.

And so $R(\alpha) \subset R(\alpha + 1) = P(R(\alpha))$ since $R(\alpha) \in P(R(\alpha))$, and now $\alpha \leq \beta \Rightarrow R(\alpha) \subseteq R(\beta)$ follows by an easy induction.

\end{proof}
If $x \in R(\gamma)$, then there is a least $\beta$ such that $x \in R(\beta)$, and this $\beta$ must be a successor ordinal, $\beta = \alpha + 1$. We define the \emph{rank} of $x$ to be that $\alpha$. Thus

\begin{itemize}%
\item $\emptyset$ has rank $0$,


\item $1 \coloneqq \{\emptyset\}$ has rank $1$,


\item $\{1\}$ and $2 \coloneqq \{0, 1\}$ have rank $2$,



\end{itemize}
and so on. Each ordinal $\alpha$ has rank $\alpha$.

\begin{theorem}
\label{}\hypertarget{}{}
For every element $x$ of $V$, there is some $\alpha \in On(V)$ such that $x \in R(\alpha)$. Thus every set $x$ appears as an element somewhere within the cumulative hierarchy

\begin{displaymath}
R(0) \subset R(1) \subset \ldots \subset R(\omega) \subset R(\omega + 1) \subset \ldots \subset R(\omega + \omega) \subset \ldots
\end{displaymath}
\end{theorem}
The proof is essentially that $\bigcup_{\alpha \in On(V)} R(\alpha)$ is an $\in$-[[well-founded relation|inductive set]] of $V$, and so must be all of $V$ since $(V, \in)$ is well-founded (by the axiom of foundation). Details may be found in any reasonable text on ZFC set theory, for example \hyperlink{Kunen}{Kunen}.

\begin{remark}
\label{}\hypertarget{}{}
The notation $V$ so widely seen in set theory texts and articles is a kind of visual pun that refers to the cumulative hierarchy: one imagines the V as outlining an angle, with a horizontal cross-section of the space inside the angle at height $\alpha$ suggesting a set $R(\alpha)$, which expands as $\alpha$ increases; the ordinals $\alpha$ themselves may be pictured as vertebrae of a spine or line therein. All sets in the cumulative hierarchy lie somewhere within the V.

\end{remark}
\hypertarget{in_algebraic_set_theory}{}\subsection*{{In algebraic set theory}}\label{in_algebraic_set_theory}

The idea of the cumulative hierarchy is realized in [[algebraic set theory]] via the construction of an initial ``ZF-algebra''. In broad-brush terms, there is a general connection between well-foundedness and initial algebras, as in Paul Taylor's theory of recursion where an initial $T$-algebra (for a [[taut functor]] $T$) is seen as a [[well-founded coalgebra]] $(X, \theta: X \to T X)$ for which $\theta$ is an isomorphism (i.e., a maximal well-founded coalgebra). Algebraic set theory can be seen as exploiting this connection and working out the details in cases specific to operations on ``small sets'', eventually enabling one to get at the cumulative hierarchy \emph{per se}, i.e., the universe of \emph{well-founded sets} (as well as the universe of ordinals, etc.).

This deserves to be discussed at length, but let us try to give a few hints for now. One starts with a [[pretopos]] $\mathcal{C}$ (whose objects are regarded as ``classes'') equipped with a suitable notion of ``smallness'': to say a map $f: E \to X$ in the pretopos is ``small'' means intuitively that all its [[fibers]] are ``small'' (i.e., sets). Thus one assumes some reasonable axioms on the class of small maps in $\mathcal{C}$, including the existence of a universal small map ``$el$'': $E \to U$, with the elements $u$ of $U$ naming small sets and the fiber over $u$ the actual (small) set of its elements. The smallness axioms allow one to construct a small-[[power set]] functor $P_s: \mathcal{C} \to \mathcal{C}$; intuitively this sends a class $C$ to the class of small subsets of $C$. This carries a [[monad]] structure whose algebras $(X, \sup: P_s X \to X)$ are ``small-complete'' posets in $\mathcal{C}$.

To get at the actual cumulative hierarchy (with attendant global membership relation $\in$), one defines a \emph{ZF-algebra} to be a small-complete poset $(V, \leq)$ in $\mathcal{C}$, equipped with a function $s: V \to V$ satisfying suitable conditions; here one is to think of $\leq$ as ``inclusion'' and $s(x)$ as a singleton $\{x\}$. The relation $x \in y$ can then be interpreted as $s(x) \leq y$. The initial object in the category of ZF-algebras then captures the desired universe of well-founded sets.

\begin{remark}
\label{}\hypertarget{}{}
The details of the construction of the initial ZF-algebra should be examined with attention to connections with Taylor's theory of well-founded coalgebras; for example, systematic use is made of bisimulations which has a general meaning in coalgebra theory.

\end{remark}
This program, initiated by [[André Joyal]] and [[Ieke Moerdijk]], permits a fine-grained analysis of \emph{intuitionistic} ZF-set theory and intuitionistic ordinals. The reader is referred to their \hyperlink{JM}{monograph} for details, and to the \href{http://www.phil.cmu.edu/projects/ast/}{Algebraic Set Theory} page for further pointers to the literature.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[constructible universe]]


\item [[reflection principle]]


\item [[algebraic set theory]]


\item [[set-theoretic multiverse]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The cumulative hierarchy made its first appearance in

\begin{itemize}%
\item D. Mirimanoff, \emph{Les antinomies de Russell et de Burali-Forti et le probl\`e{}me fondamental de la th\'e{}orie des ensembles} , L'enseignement Math\'e{}matique \textbf{19} (1917) pp.37-52. (\href{http://retro.seals.ch/cntmng?pid=ensmat-001:1917:19::9}{pdf}; 19,4MB)

\end{itemize}
Another historically important contribution is

\begin{itemize}%
\item Dana Scott, \emph{Axiomatizing Set Theory} , pp.207-214 in \emph{Axiomatic Set Theory} - Proc. Symp. Pure Math. \textbf{13}, AMS Providence 1974.

\end{itemize}
A modern textbook account can be found e.g. in

\begin{itemize}%
\item Kenneth Kunen, \emph{Set Theory: An Introduction to Independence Proofs}, Studies in Logic and the Foundations of Mathematics Vol. 102 (2006), Elsevier.

\end{itemize}
For algebraic set theory consult the following monograph

\begin{itemize}%
\item Andr\'e{} Joyal and Ieke Moerdijk, \emph{Algebraic Set Theory}, London Math. Soc. Lecture Series Notes 220 (1995), Cambridge University Press.

\end{itemize}


\end{document}