Transitive set

In set theory, a set (or class) A is transitive, if

• whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,
• whenever x ∈ A, and x is not an urelement, then x is a subset of A.

The transitive closure of a set A is the smallest (with respect to inclusion) transitive set B which contains A. Suppose one is given a set X, then the transitive closure of X is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \bigcup \{ X, \bigcup X, \bigcup \bigcup X, \bigcup \bigcup \bigcup X, \bigcup \bigcup \bigcup \bigcup X, ... \}

.

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.

An ordinal number may be defined as a transitive set whose members are also transitive.

A set, X, is transitive if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): \bigcup X \subseteq X.

A set, X, which does not contain urelements is transitive if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): X \subset \mathcal{P}(X).

Transitive classes

Similarly, a class M is transitive if every element of M is a subset of M.

Transitive models of set theory

A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system. Transitivity is an important factor in determining the absoluteness of formulas.

See also

• Union (set theory)
• Power set

cs:Tranzitivní třída

fr:Ensemble transitif