Transitive set

In set theory, a set A is called transitive, if

• Follows from and getting that,

Or equivalently if

• Each element of A is a subset of A.

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

Examples

• An ordinal as defined by John von Neumann is a transitive set with the property that each element is transitive again.
• A Grothendieck universe is a transitive set by definition.
• Transitive classes are used as models for set theory itself.

Properties

• A lot of X is transitive if, with the union of all elements of X is so.
• If X is transitive, then it is also transitive.
• When X and Y are transitive quantities, then transitive
• In general, if X is a class whose elements are all transitive subsets, then a transitive class.
• A set X is transitive if and only if X is a subset of the power set of X.
• The power set of a transitive set is transitive again. This property is used to view in the von Neumann hierarchy, that all levels of the hierarchy are transitive.