Bob Mayans wrote: << ... This is the so-called epsilon induction, and it requires not the Axiom of Choice but the Axiom of Restriction. Suppose that for any set A, if P(a) holds for each a in A, then P(A) holds. Then P(A) holds for all sets...

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I wrote: << I think we'd also have to suppose P(Ø) to conclude P(A) holds for all sets. Michael Kleber wrote: << Nope! It's the base-case-free induction. ... I remember learning that trick from Gerald Sacks as an undergraduate, and walking around in a delighted haze at the cleverness for the rest of the day...

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Thanks, Michael (and Bob M. via private e-mail) for setting me straight. Unfortunately, I managed to be in a haze *before* I grasped this incredibly clever kind of induction. --Dan