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Sniffnoy comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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One thing bothering me -- is there any way to define a well-founded set without using infinitary reasoning? It's easy enough to say that all sets are well-founded without it, by just stating that ∈ is well-founded -- I mean, that's what the standard axiom of foundation does, though with the classical definition -- but in contexts where that doesn't hold, you need to be able to distinguish a well-founded set from an ill-founded one. Obvious thing to do would be to take the transitive closure of the set and ask if ∈ is well-founded on that, but what bugs me is that constructing the transitive closure requires infinitary reasoning as well. Is there something I'm missing here?
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.