I think I finally have a good way of explaining Russel’s paradox.
Think of the box that holds all of the boxes.
This box is the biggest possible box. This seems to work.
The definition of a box is to contain something smaller than itself.
What if instead of the word box, I replace it with its definition: Something that holds things smaller than itself. If we want to have THE thing which holds ALL things smaller than its, then it would have to be smaller than itself and bigger than itself at the same time. As in, The Biggest Box, as that which can hold/contain things, must hold itself, but since this is impossible we cannot conclude that this Biggest box exists. There are only some Boxes.
Or, think of The Container that contains All the containers. If ALL the containers are contained in this One Container, then this container should also be contained in this One Container. This One Container that supposedly contains All containers can no longer said to be a container. By its definition/property (containment) then becomes contradictory. How can a container contain itself? It cannot. Can’t we just change the name to, say, The ONE?
No. that is, not if we are trying to provide the foundation for what exists based on containment (set), which is a collection of objects determined by a property. The sole “being” of the set is determined by how the objects relate to each other in the set.
It’s tempting to give a Wittgensteinian answer: “When we try to determine what things are based on their definition and not by their use, we run into all sorts of funny trouble.” This would not hold up for a number of reasons. First, though sets are defined by their composition, “use” is not excluded and is in fact an essential part in identifying a set. A set is determined by a property, which is chosen based on its use. Badiou’s “Count-as-one” is essentially the same as Wittgenstein’s “use” or “game.” Second, if a being is determined by its use, what would it look like to determine “all that exists” by its “use”? We would have to remain silent on this question, or have many different definitions. Third, if we are looking to define beings with use, then we would have to determine how we use, use, in order to make any attempt to define “all that exists.” Finally, for those that are skeptical in applying set-theory to ontology, and therefore skeptical about rejecting The One, the being that contains all beings, one would have to demonstrate an alternate formulation of differentiating between beings, that is, the one and the multiple.
If The One is defined as that which contains everything, then it is contradictory and another way of making distinctions between beings must be proposed, that is, an alternative ontology.
What do you think?