Definition:Strictly Well-Founded Relation/Definition 1

Definition

Let $\struct {S, \RR}$ be a relational structure.

$\RR$ is a strictly well-founded relation on $S$ if and only if every non-empty subset of $S$ has a strictly minimal element under $\RR$.

Also known as

A strictly well-founded relation is also known in the literature as a foundational relation.

It is commonplace in the literature and on the internet to use the term well-founded relation for strictly well-founded relation.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the more cumbersome and arguably more precise strictly well-founded relation in preference to all others.

Some sources do not hyphenate, and present the name as strictly wellfounded relation.

Also see

• Equivalence of Definitions of Strictly Well-Founded Relation

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.21$