# Set element of itself

#### 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?

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#### 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

Science Advisor
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.)

#### 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

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

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