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another important axiom of set theory
The '''Axiom of Foundation''' is the most recent of the [[axiom]]s of [[Zermelo-Fraenkel]] set theory to be added to the list. It states that for every set ''A'' the sequence:
* ''A''
* elements of ''A''
* elemenents of elements of ''A''
* elemenets of elements of elemeents of ''A''
* etc.
must eventually stop. In other words, every set ''A'' has a bottom, or foundation. It is therefore impossible in Zermelo-Fraenkel set theory to define a set as ''A={A}''. In other words, ''A'' cannot be defined to be an element of itself.
This axiom eliminates [[circular argument|circular definitions]] from the logic of set theory and also quashes several [[paradox]]es, a great achievement for [[logic]].
[[Category:Logic]]
* ''A''
* elements of ''A''
* elemenents of elements of ''A''
* elemenets of elements of elemeents of ''A''
* etc.
must eventually stop. In other words, every set ''A'' has a bottom, or foundation. It is therefore impossible in Zermelo-Fraenkel set theory to define a set as ''A={A}''. In other words, ''A'' cannot be defined to be an element of itself.
This axiom eliminates [[circular argument|circular definitions]] from the logic of set theory and also quashes several [[paradox]]es, a great achievement for [[logic]].
[[Category:Logic]]