Set A: Element of Itself? Meaning Explained

[ronaldor9]

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?

 

[Dragonfall]

Yes, if you assume it to be an axiom. See http://en.wikipedia.org/wiki/Non-well-founded_set

If you don't assume it or its negation, you cannot prove this either way.

 

[ronaldor9]

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]

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.)

 

[Dragonfall]

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]

Oops, I flipped that one mentally to "{x: x is an element of x}" which is the empty set with Foundation.