Axiom:Axiom of Foundation
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Axiom
For all non-empty sets, there is an element of the set that shares no element with the set.
That is:
- $\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$
The antecedent states that $S$ is not empty.
Also defined as
It can also be stated as:
- For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint.
- A set contains no infinitely descending (membership) sequence.
- A set contains a (membership) minimal element.
- The membership relation is a strictly well-founded relation on any non-empty set.
Also known as
The axiom of foundation is also known as the axiom of regularity.
Sources
- 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF9}$
- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html