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In "set theory, a "set A is called transitive if either of the following equivalent conditions hold:

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



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 Vα and Lα leading to the construction of the "von Neumann universe V and "Gödel's constructible universe L are transitive sets. The "universes L and V themselves are transitive classes.

This is a complete list of all finite transitive sets with up to 20 brackets:[1]


A set X is transitive if and only if , where is the "union of all elements of X that are sets, . If X is transitive, then is transitive. If X and Y are transitive, then XY∪{X,Y} is transitive. In general, if X is a class all of whose elements are transitive sets, then is transitive.

A set X which does not contain urelements is transitive if and only if it is a subset of its own "power set, The power set of a transitive set without urelements is transitive.

Transitive closure[edit]

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

Note that this is the set of all of the objects related to X by the "transitive closure of the membership relation.

Transitive models of set theory[edit]

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. 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.["clarification needed][2]

See also[edit]


External links[edit]

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