is a transitive closure of iff is an intersection of all transitive supersets of (that is, every transitive superset of contains as a subset). Transitive models of set theory 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. 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. See also End extension Transitive relation Supertransitive class References Set theory