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, $\map P S = S \in S$ is a property on which we can use the comprehension principle to build this set:


 * $T = \set {S: S \in S}$

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

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


 * $R = \set {S: S \notin S}$

($R$ for, 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.

Also presented as
This result can also be presented as:
 * There is no set $A$ that has every set as its element.

Its proof follows the same lines: by assuming that such an $A$ exists, and considering the set $\set {x \in A: \map R x}$ where $\map R x$ is the property $x \notin x$.

The same conclusion is reached.

Also known as
This result is also known as Russell's antinomy.

Also see

 * Barber Paradox