Organization:Foundational relation

In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimal element.

Formally, let (A, R) be a binary relation structure, where A is a class (set or proper class), and R is a binary relation defined on A. Then (A, R) is a foundational relation if any nonempty subset in A has an R-minimal element. In predicate logic,

$(\mathrm{\forall }S)(S\subseteq A\wedge S=\mathrm{\varnothing }\Rightarrow (\mathrm{\exists }x\in S)(\mathrm{\forall }y\in S)(\mathrm{\neg }yRx)),$

in which $\mathrm{\varnothing }$ denotes the empty set. Here $x$ is an R-minimal element in the subset S, since none of its R-predecessors is in S.

See also

• Binary relation
• Well-order