If there is a category "Empty Set" it has to have the property "emptiness". It must have this property that separates it from every other set. Thus it is not propertyless - contradiction?
I don't see a contradiction here any more than I did back at Question 26649 , which is nearly identical. Yes, the empty set has the property of being empty and is the only set having that property. But the emptiness of the empty set doesn't imply that the empty set has no properties. On the contrary, it has the property of being empty, being a set, being an abstract object, being distinct from Mars, being referred to in this answer, etc. Why would anyone think that the empty set must lack all properties?