lew (02/08/83)
Thanks to Yigal Arens and Jerry Leichter for their rapid and informative
responses. I was confusing the substitution of y into a formula containing
x as a free variable, with a formula containing y as a free variable.
Jerry is right about the Monk book, he defines a set as "a class small
enough to be a member of some other class." He defines a universal
class, not a universal set.
I think it is interesting that the ZF formulation still lets us define
y={x: x<z ^ x~<x}. With z=y , when we come to the question of y's self-
membership we have y<y <-> y<y ^ y ~< y. Clearly the right side is false,
so y ~< y. This axiom just breaks Russell's paradox in favor of y ~< y.
We get away with "the set of all non-self-members except itself".
Lew Mammel, Jr. ihuxr!lew