In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, a hereditary set (or pure set) is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
whose elements are all hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on.
Examples
For example, it is
vacuously true
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied.
It is sometimes said that a s ...
that the empty set is a hereditary set, and thus the set
containing only the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
is a hereditary set. Similarly, a set
that contains two elements: the empty set and the set that contains only the empty set, is a hereditary set.
In formulations of set theory
In formulations of set theory that are intended to be interpreted in the
von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory ( ...
or to express the content of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
, ''all'' sets are hereditary, because the only sort of object that is even a candidate to be an element of a set is another set. Thus the notion of hereditary set is interesting only in a context in which there may be
urelement
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set (has no elements), but that may be an element of a set. It is also referred to as an atom or individ ...
s.
Assumptions
The inductive definition of hereditary sets presupposes that set membership is
well-founded
In mathematics, a binary relation is called well-founded (or wellfounded or foundational) on a set (mathematics), set or, more generally, a Class (set theory), class if every non-empty subset has a minimal element with respect to ; that is, t ...
(i.e., the
axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoin ...
), otherwise the recurrence may not have a unique solution. However, it can be restated non-inductively as follows: a set is hereditary if and only if its
transitive closure
In mathematics, the transitive closure of a homogeneous binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be ...
contains only sets.
In this way the concept of hereditary sets can also be extended to
non-well-founded set theories in which sets can be members of themselves. For example, a set that contains only itself is a hereditary set.
See also
*
Hereditarily countable set
In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets.
Results
The inductive definition above is well-founded and can be expressed in the language of first-order set theory.
Equivalent p ...
*
Hereditarily finite set
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 t ...
*
Well-founded set
References
*
{{Mathematical logic
Set theory