Transitive set

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 V_α and L_α 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:

A set X is transitive if and only if ${\displaystyle \bigcup X\subseteq X}$, where ${\displaystyle \bigcup X}$ is the union of all elements of X that are sets, ${\displaystyle \bigcup X=\{y\mid (\exists x\in X)y\in x\}}$. If X is transitive, then ${\displaystyle \bigcup X}$ is transitive. If X and Y are transitive, then X∪Y∪{X,Y} is transitive. In general, if X is a class all of whose elements are transitive sets, then ${\displaystyle X\cup \bigcup X}$ is transitive.

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