In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set.

A recursive definition of well-founded hereditarily finite sets is as follows:

Only sets that can be built by a finite number of applications of these two rules are hereditarily finite.

This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:

In this way, the number of sets with ${\displaystyle n}$ bracket pairs is

The set ${\displaystyle \{\{\},\{\{\{\}\}\}\}}$ is an example for such a hereditarily finite set and so is the empty set ${\displaystyle \{\}}$, as noted. On the other hand, the sets ${\displaystyle \{7,{\mathbb {N} },\pi \}}$ or ${\displaystyle \{3,\{{\mathbb {N} }\}\}}$ are examples of finite sets that are not hereditarily finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when ${\displaystyle {\mathbb {N} }=\{0,1,2,\dots \}}$.

The class of all hereditarily finite sets is denoted by ${\displaystyle H_{\aleph _{0}}}$, meaning that the cardinality of each member is smaller than ${\displaystyle \aleph _{0}}$. (Analogously, the class of hereditarily countable sets is denoted by ${\displaystyle H_{\aleph _{1}}}$.) It can also be denoted by ${\displaystyle V_{\omega }}$, which denotes the ${\displaystyle \omega }$th stage of the von Neumann universe.

${\displaystyle H_{\aleph _{0}}}$ is in bijective correspondence with ${\displaystyle \aleph _{0}}$. A theory which proves it to be a set also proves it to be countable.