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.

He discovered it in 1901.