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...
>>

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...
>>

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