In set theory, a branch of mathematics, a set
is called transitive if either of the following equivalent conditions holds: Similarly, a class
Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals).
leading to the construction of the von Neumann universe
and Gödel's constructible universe
This is a complete list of all finite transitive sets with up to 20 brackets:[1] A set
is the union of all elements of
is a class all of whose elements are transitive sets, then
(The first sentence in this paragraph is the case of
that does not contain urelements is transitive if and only if it is a subset of its own power set,
The power set of a transitive set without urelements is transitive.
The transitive closure of a set
is the smallest (with respect to inclusion) transitive set that includes
is a transitive set including
We prove by induction that
: The base case holds since
Note that this is the set of all of the objects related to
by the transitive closure of the membership relation, since the union of a set can be expressed in terms of the relative product of the membership relation with itself.
The transitive closure of a set can be expressed by a first-order formula:
is an intersection of all transitive supersets of
Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models.
The reason is that properties defined by bounded formulas are absolute for transitive classes.
[3] A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system (provided that the element relation of the model is the restriction of the true element relation to the universe of the model).
Transitivity is an important factor in determining the absoluteness of formulas.
In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity.
is defined to be strongly transitive if, for each set
, there exists a transitive superset
A strongly transitive class is automatically transitive.
This strengthened transitivity assumption allows one to conclude, for instance, that
contains the domain of every binary relation in