Transitive set

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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:

.

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

A set, X, which does not contain urelements is transitive if and only if

[edit] Transitive classes

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

[edit] See also

• Union (set theory)
• Power set