Definition:Foundational Relation

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\left({A, \mathcal R}\right)$ be a relational structure where $A$ is either a proper class or a set.


Then $\mathcal R$ is a foundational relation on $A$ if and only if every non-empty subset of $A$ has an $\mathcal R$-minimal element.


That is, $\mathcal R$ is a foundational relation on $A$ if and only if:

$\forall s: \left({s \subseteq A \land s \ne \varnothing}\right) \implies \exists y \in s: \forall z \in s: \neg \left({z \mathrel{\mathcal R} y}\right)$

where $\varnothing$ is the empty set.


Also see


Special case


Sources