Difference between revisions of "Axiom of Foundation"
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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: | 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'' | * ''A'' | ||
| − | * | + | * ''A<sub>1</sub>'' an element of ''A'' |
| − | * | + | * ''A<sub>2</sub>'' an element of ''A<sub>1</sub>'' |
| − | * | + | * ''A<sub>3</sub>'' an element of ''A<sub>2</sub>'' |
* etc. | * etc. | ||
Latest revision as of 01:54, April 12, 2010
The Axiom of Foundation is the most recent of the axioms of Zermelo-Fraenkel set theory to be added to the list. It states that for every set A the sequence:
- A
- A1 an element of A
- A2 an element of A1
- A3 an element of A2
- 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 definitions from the logic of set theory and also quashes several paradoxes, a great achievement for logic.