Sets are collections of things. Under **Unrestricted Comprehension**, a set can be made with any property at all. For example, with the property of being a cat, you can have the set of cats: the set that has as its members anything that has the property of being a cat.

You can even evaluate the set itself with the property it is the set of. The set of cats is not a cat. And so, the set of cats is not a member of itself. What about the set of non-cats, of those things that are not a cat? The set of non-cats, the collection of all those things that aren’t cats, is itself not a cat. And so, the set of non-cats has itself as a member.

Some sets (the set of cats) do not include itself as a member while others (the set of non-cats) do. Russell’s paradoxical set is:

**Let R = { x | x ∉ x }**

This says: let set *R* be the set of anything *x* that is not a member of itself.

The set of cats is a member of *R*, since the set of cats has the property of non-membership of itself that is specified by *R*, since it’s not a member of itself. On the other hand, the set of non-cats is not a member of *R*, since the set of non-cats includes itself as a member, itself being not a cat, and thus does not have the property specified by *R*. Russell’s paradoxical query goes: what about *R*, does *R* have itself as a member or not?

Consider that *R* has itself as a member. However, *R* specifies sets that do not have themselves as a member. And so, *R* cannot be a member of itself. But maybe that’s the answer, and we made a discovery: *R* is not a member of itself! Unfortunately no. In this case, *R* must be a member of itself, since this is exactly what R specifies. The result is that if set *R* is a member of itself, then set *R* is not a member of itself, and if set *R* is not a member of itself, then set *R* is a member of itself; i.e. *R* ∈ *R* ⇔ *R* ∉ *R*. This result of Russell’s Paradox can be written:

**Let R = { x | x ∉ x }, then R ∈ R ⇔ R ∉ R.**

Combine this result with the truth of the Principle of Bivalence of Sets that *R* either is or is not a member of itself, then the contradiction that *R* both is and is not a member of itself is readily derivable. (*R* ∈ *R* & *R* ∉ *R*) follows from the mere construction of set *R* and the truth that *R* ∈ *R* v *R* ∉ *R*.

Historically, set theory has been amended in various ways, typically through axioms, so that *R* cannot be stated. I take it that all such solutions are similar for doing away with Unrestricted Comprehension, restricting how sets can be built from properties.