Wikipedia / Transitive set
Wikipedia / Transitive set
In set theory, a branch of mathematics, a set is called transitive if either of the following equivalent conditions holds:
Similarly, a class.
Examples
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). The class of all ordinals is a transitive class.
leading to the construction of the von Neumann universe and Gödel's constructible universe are transitive sets. The universes themselves are transitive classes.
This is a complete list of all finite transitive sets with up to 20 pairs of brackets:
Properties
is the union that are sets; formally,
.
is transitive.
.)
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.
Transitive closure
The transitive closure. can be equivalently defined as
Proof. Then we claim that the set
.
is transitive.
. This completes the proof.
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.
Transitive models of set theory
Transitive classes are often used for construction of interpretationss of set theory in itself, usually called inner models. The reason is that properties defined by bounded formulas are absolute for transitive classes.
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 contains the domain of every binary relation.
See also
Notes
References