If there is a category "Empty Set" it has to have the property "nothingness". Thus it is not propertyless - contradiction?

As far as I can see, the definitive property of the empty set is not nothingness but instead emptiness: It's the one and only set having (containing, possessing) no members at all. The empty set can be empty, in that sense, without itself being nothing. So I see no threat of contradiction here. Indeed, the empty set can belong to a non-empty set, such as the set { { } } , which couldn't happen if the empty set were nothing.

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