I know one way; it cannot be stated in ZFC↺ (ZFC without foundation), but it can be stated in MK↺ (the Morse–Kelley class theory version): a set is well-founded iff it belongs to every transitive class of sets (that is every class K such that x ∈ K whenever x ⊆ K); it is immediate that we may prove properties of these sets by induction on membership, and a set is well-founded if all of its elements are, so this is a correct definition. However, it requires quantification over all classes (not just sets) to state.