Smullyan and Fitting say that a class $A$ is well-founded iff every non-empty subclass $B \subseteq A$ has an element $x$ such that $x \cap B = \varnothing$.
In the presence of the axioms of infinity and replacement (I don't see a way around them anyway), every set has a transitive closure, so requiring the above to hold only when $B$ is a set gives the same result.
Wikipedia gives a much stronger definition, defining a set as well-founded iff its transitive closure is well-founded (in the above sense).
In an intuitionistic setting, non-empty sets/classes having $\epsilon$-minimal elements does not permit well-founded induction, so apparently there's an intuitionistic notion that explicitly permits induction (which can have similar strong and weak forms as above).
In the matter of well-founded relations, all the above considerations continue to hold, but the equivalence of the first two notions then seems to depend both on the axiom of infinity and the axiom of foundation.
So is there any set of terminology for talking about this zoo of notions?
