Definition:Well-Founded Relation/Class Theory

Definition

Let $A$ be a class.

Let $\RR$ be a relation on $A$.

$\RR$ is a well-founded relation on $A$ if and only if:

every non-empty subclass $A$ has an initial element.

Also known as

The term well-founded relation is often used the literature for what on $\mathsf{Pr} \infty \mathsf{fWiki}$ we call a strictly well-founded relation.

In order to emphasise the differences between the two, at some point a $\mathsf{Pr} \infty \mathsf{fWiki}$ editor coined the term strongly well-founded relation.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the less unwieldy term well-founded relation in preference to others.

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

Also see

• Results about well-founded relations can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.): Chapter $10$: Mostowski-Shepherdson Mappings: $\S 1$ Relational systems: Well-foundedness