Transitive set
In set theory , a set is called transitive if
- from and it always follows that , in signs:
- ,
or equivalent if
- every element of that is a set is a subset of .
'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 .
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 .
- If is transitive, then is also transitive.
- If and are transitive sets, then is also transitive
- In general, if is a class whose elements are all transitive sets, then is a transitive class.
- A set is transitive if and only if is a subset of the power set of .
- 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:
- .
In this case , the above definition results as a special case.
Remarks
- ↑ Only elements that are sets go into this union, i.e. no ('real') primordial elements.
- ↑ Wolfram Pohlers: Set theory (PDF) , University of Münster, Institute for Mathematical Logic and Basic Research, lecture notes, SS 1994, page 31
See also
literature
- Thomas Jech [originally published in 1973]: The Axiom of Choice . Dover Publications, 2008, ISBN 0-486-46624-8 .