axiom of foundation
The axiom of foundation (also called the axiom of regularity) is an axiom of ZF set theory prohibiting circular sets and sets with infinite levels of containment. Intuitively, it that every set can be built up from the empty set. There are several equivalent formulations, for instance:
For any nonempty set there is some such that .
For any formula , if there is any set such that then there is some such that but there is no such that .
One important consequence of this property is that no set can contain itself. For instance, if there were a set such that then we could define a function for all , which would then have the property that for all .
|Title||axiom of foundation|
|Date of creation||2013-03-22 13:04:31|
|Last modified on||2013-03-22 13:04:31|
|Last modified by||Henry (455)|
|Synonym||axiom of regularity|