Transitive set
In set theory, a branch of mathematics, a set formula_1 is called transitive if either of the following equivalent conditions hold: Similarly, a class formula_9 is transitive if every element of formula_9 is a subset of formula_9. 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. Any of the stages formula_12 and formula_13 leading to the construction of the von Neumann universe formula_14  and Gödel's constructible universe formula_15 are transitive sets. The universes formula_14 and formula_15 themselves are transitive classes. This is a complete list of all finite transitive sets with up to 20 brackets: Properties. A set formula_65 is transitive if and only if formula_66, where formula_67 is the union of all elements of formula_68 that are sets, formula_69. If formula_65 is transitive, then formula_67 is transitive. If formula_65 and formula_73 are transitive, then formula_74 and formula_75 are transitive. In general, if formula_76 is a class all of whose elements are transitive sets, then formula_77 and formula_78 are transitive. (The first  sentence in this paragraph is the case of formula_79.) A set formula_65 that does not contain urelements is transitive if and only if it is a subset of its own power set, formula_81 The power set of a transitive set without urelements is transitive. Transitive closure. The transitive closure of a set formula_65 is the smallest (with respect to inclusion) transitive set that includes formula_65. Suppose one is given a set formula_65, then the transitive closure of formula_65 is Proof. Denote formula_87 and formula_88. Then we claim that the set is transitive, and whenever formula_90 is a transitive set including  formula_91 then formula_92. Assume formula_93. Then formula_94 for some formula_95 and so formula_96. Since formula_97, formula_98. Thus formula_99 is transitive. Now let formula_90 be as above. We prove by induction that formula_101 for all formula_102, thus proving that formula_92: The base case holds since formula_104. Now assume formula_101. Then formula_106. But formula_90 is transitive so formula_108, hence formula_109. This completes the proof. Note that this is the set of all of the objects related to formula_65 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 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.