In
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
, a hereditary set (or pure set) is a
set 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. For example, the statement "she d ...
that the empty set is a hereditary set, and thus the set
containing only the
empty set 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 (ZF ...
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 such ...
, ''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, but that may be an element of a set. It is also referred to as an atom or individual.
Theory
There ...
s.
Assumptions
The inductive definition of hereditary sets presupposes that set membership is
well-founded (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 non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the ...
), 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 binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite ...
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
*
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