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

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CRGreathouse

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{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.)

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CRGreathouse

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