Russell's Paradox

Theorem
The comprehension principle leads to a contradiction.

Proof
Sets have elements.

Some of those elements may themselves be sets.

So, given two sets $$S$$ and $$T$$, we can ask the question: Is $$S$$ an element of $$T$$? The answer will either be "yes" or "no".

In particular, given any set $$S$$, we can ask the question: Is $$S$$ an element of $$S$$? Again, the answer will either be "yes" or "no".

Thus, $$P \left({S}\right) = S \in S$$ is a property on which we can use the comprehension principle to build this set:

$$T = \left\{{S: S \in S}\right\}$$

... which is the set of all sets which contain themselves.

Alternatively, we can apply the comprehension principle to build this set:

$$R = \left\{{S: S \notin S}\right\}$$

(R for Russell, of course.)

We ask the question: Is $$R$$ itself an element of $$R$$?

There are two possible answers: "yes" or "no".

If $$R \in R$$, then $$R$$ must satisfy the property that $$R \notin R$$, so from that contradiction we know that $$R \in R$$ does not hold.

So the only other answer, $$R \notin R$$, must hold instead. But now we see that $$R$$ satisfies the conditions of the property that $$R \in R$$, so we can see that $$R \notin R$$ doesn't hold either.

Thus we have generated a contradiction from the comprehension principle.

Comment
This paradox is an antinomy arising from the inherent contradiction in allowing a set to contain itself.

The paradox is overcome by disallowing that possibility.

The system ZFC of axiomatic set theory implements this constraint by means of the Axiom of Foundation.