13 Feb
2003
13 Feb
'03
10:17 p.m.
<< 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. The theories of transfinite induction and well-founded relations both generalize induction; see Thomas Jech's book on set theory.
I think we'd also have to suppose P(Ø) to conclude P(A) holds for all sets. Otherwise we could prove any P that's always false holds for all sets. --Dan