# Set element of itself

Can a set A be an element of A, or can A be not an element of A? And what would such mean in plain-speak?

Last edited:

thanks! By the way, why is it that {x: x=x} and {x: x not an element of x} do not constitute a set? The latter I would think would constitute the null set, but apparently this is wrong.

CRGreathouse
Homework Helper
thanks! By the way, why is it that {x: x=x} and {x: x not an element of x} do not constitute a set? The latter I would think would constitute the null set, but apparently this is wrong.

{x: x = x} is a proper class.

I would have thought that, with the Axiom of Foundation, {x: x is not an element of x} would be the empty set. (Without it might be too big to be a set, and can't be proven to be empty.)

With foundation, {x:x is not an element of x} is the proper class V. In naive set theory it forms the Russel paradox.

CRGreathouse