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 and only if any nonempty subset in A has a R-minimal element. In predicate logic,

in which ∅ denotes the empty set, and R⁻¹{x} denotes the class of the elements that precede x in the relation R. That is,

Here x is an R-minimal element in the subset S, since none of its R-predecessors is in S.