Definition:Strongly Well-Founded Relation
2 Also known as
3 Remarks
4 Also see
4.1 Weaker properties
Let $A$ be a class.
Let $\mathcal R$ be a relation on $A$.
Then $A$ is strongly well-founded if and only if:
whenever $B$ is a non-empty subclass of $A$, $B$ has an $\mathcal R$-minimal element.
Most sources that define this concept simply call it well-founded. However, that term is most commonly defined in a weaker sense.
Thus on $\mathsf{Pr} \infty \mathsf{fWiki}$ we have invented the term "strongly well-founded" to avoid ambiguity.
This article, or a section of it, needs explaining, namely:
The above link is to a page which says nothing about well-foundedness. On the other hand there is also a page Definition:Well-Founded and another Definition:Well-Founded Set. Methinks that, this particular can of worms having been opened, an attempt should be made by its opener to arrange the contents into neat rows.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
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According to Well-Founded Relation Determines Minimal Elements, the Axiom of Foundation implies that every foundational relation is strongly well-founded.
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As the definition quantifies over general subclasses, strong well-foundedness is not a first-order property. Thus it cannot be used in a formula of a theory, like NBG class-set theory, that only allow quantification over sets; for example it cannot be used with the axiom schema of specification in NBG set theory, and statements using it must generally be framed as schemas. A higher-order theory such as Morse-Kelley set theory allows it to be used more freely and directly.
A concrete example:
Let $A$ be a class, and let $\mathcal R$ be a relation on $A$.
One cannot directly use the axiom schema of specification in NBG theory to form the class of all subsets $x$ of $A$ such that the restriction of $\mathcal R$ to $A \setminus x$ is a strongly well-founded relation.
Weaker properties
Definition:Foundational Relation
2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.): $\S 10.1$
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Definitions/Relation Theory
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