Transitive set

In set theory , a set is called transitive if ${\displaystyle A}$

• from and it always follows that , in signs: ${\displaystyle x\in A}$ ${\displaystyle y\in x}$ ${\displaystyle y\in A}$

${\displaystyle \forall x,y:\,x\in A\land y\in x\Rightarrow y\in A}$,

or equivalent if

• every element of that is a set is a subset of . ${\displaystyle A}$ ${\displaystyle A}$

'Real' (i.e. different from the empty set ) primordial elements are not important here.
Similarly, a class is called transitive if every element of is a subset of . ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$

Examples

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

properties

• A set is transitive if and only if , where is the union of all elements of . ${\displaystyle A}$ ${\displaystyle \bigcup A\subseteq A}$ ${\displaystyle \bigcup A=\bigcup _{x\in A}x=\{y|(\exists x\in A)y\in x\}=\{y|y\in ^{2}A\}}$ ${\displaystyle A}$
• If is transitive, then is also transitive. ${\displaystyle A}$ ${\displaystyle \bigcup A}$
• If and are transitive sets, then is also transitive ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\cup B\cup \{A,B\}}$
• In general, if is a class whose elements are all transitive sets, then is a transitive class. ${\displaystyle A}$ ${\displaystyle A\cup \bigcup A}$
• A set is transitive if and only if is a subset of the power set of . ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$
• The power set of a transitive set is again transitive. This property is used in the Von Neumann hierarchy to understand that all levels of this hierarchy are transitive.

generalization

Given a set (or class) and a relation on it. means - transitive , if the following applies: ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R}$

${\displaystyle \forall x,y:\,x\in A\land y\,R\,x\Rightarrow y\in A}$.

In this case , the above definition results as a special case. ${\displaystyle R={\in }}$

Remarks

1. ↑ Only elements that are sets go into this union, i.e. no ('real') primordial elements.
2. ↑ Wolfram Pohlers: Set theory (PDF) , University of Münster, Institute for Mathematical Logic and Basic Research, lecture notes, SS 1994, page 31

See also

• Transitive relation
• Transitive envelope (quantity)

literature

• Thomas Jech [originally published in 1973]: The Axiom of Choice . Dover Publications, 2008, ISBN 0-486-46624-8 .